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POCKET. «24£-5-«5q 


ASIMOV ON 

NUMBERS 


"AS! MOV : S Tl IE GREATEST EXPLAINER Or THE AGE 
-CARL SAGAN. AUTHOR OF COSMOS 



ASIMOV TELLS, AS NO 
ONE ELSE CAN: 


• How to make a trillion seem small 

• Why imaginary numbers are real 

• The real size of the universe — in photons 

• How much water the world really has 

• Why googles and T-formations are larger 
than you can imagine 

• Why the zero isn’t “good for nothing” at 
all 

“With his usual aplomb, the science popularizer 
steers readers through some historic brainteasers 
and on to numerical oddities. . . . Asimov’s re- 
markable ability to simplify the complex, and his 
good-humored informality, once again transform 
potentially difficult material into a pleasurable 
reading experience.” 


— Booklist 


Books by Isaac Asimov 


Asimov on Numbers 
The Beginning and the End 
The Collapsing Universe 

Published by POCKET BOOKS 


asm 



9 

PUBLISHED BY POCKET BOOKS NEW YORK 


Dedicated to 
CATHLEEN JORDAN 
and a new beginning 



POCKET BOOKS, a Simon & Schuster division of 

GULF & WESTERN CORPORATION 

1230 Avenue of the Americas, New York, N.Y. 10020 


Copyright © 1977 by Isaac Asimov; copyright © 1959, 1960, 
1961, 1962, 1963, 1964, 1965, 1966 by Mercury Press, Inc. 


Published by arrangement with Doubleday & Company, Inc. 
Library of Congress Catalog Card Number: 76-23747 


All rights reserved, including the right to reproduce 
this book or portions thereof in any form whatsoever. 
For information address Doubleday & Company, Inc., 
245 Park Avenue, New York, N.Y. 10017 


ISBN: 0-671-44246-5 


First Pocket Books printing Octoher, 1978 
10 9 8 7 6 5 4 

POCKET and colophon are trademarks of Simon & Schuster. 


Printed in the U,S.A. 




All essays in this volume originally appeared in The Magazine 
of Fantasy and Science Fiction . Individual essays were in the 
following issues. 


Varieties of the Infinite 

A Piece of Pi 

Tools of the Trade 

The Imaginary That Isn’t 

That’s About the Size of It 

Pre-fixing It Up 

One, Ten, Buckle My Shoe 

T-Formation 

Forget It! 

Nothing Counts 
The Days of Our Years 
Begin at the Beginning 
Exclamation Point! 

Water, Water, Everywhere — 
The Proton-Reckoner 
Up and Down the Earth 
The Isles of Earth 


September 1959 
May 1960 
September 1960 
March 1961 
October 1961 
November 1962 
December 1962 
August 1963 
March 1964 
July 1964 
August 1964 
January 1965 
July 1965 
December 1965 
January 1966 
February 1966 
June 1966 



CONTENTS 


Introduction x * 

Part I NUMBERS AND COUNTING 

1. Nothing Counts 3 

2. One, Ten, Buckle My Shoe 19 

3. Exclamation Point! 36 

4. T-Formation 52 

5. Varieties of the Infinite 68 

Part II NUMBERS AND MATHEMATICS 

6. A Piece of Pi 87 

7. Tools of the Trade 101 

8. The Imaginary That Isn’t 115 

Part III NUMBERS AND 
MEASUREMENT 

9. Forget It! 131 

10. Pre-fixing It Up 147 

Part IV NUMBERS AND THE 
CALENDAR 

11. The Days of Our Years 165 

12. Begin at the Beginning 181 

Part V NUMBERS AND BIOLOGY 

13. That’s About the Size of It 199 


ix 



X 


Contents 


Part VI NUMBERS AND ASTRONOMY 


14. The Proton-Reckoner 218 

Part VII NUMBERS AND THE EARTH 

15. Water, Water, Everywhere — 231 

16. Up and Down the Earth 247 

17. The Isles of Earth 262 

Index 279 


INTRODUCTION 


Back in 1959, I began writing a monthly science 
column for The Magazine of Fantasy and Science Fiction. 
I was given carte blanche as to subject matter, approach, 
style, and everything else, and I made full use of that. I 
have used the column to range through every science in 
an informal and very personal way so that of all the writ- 
ing I do (and I do a great deal) nothing gives me so much 
pleasure as these monthly essays. 

And as though that were not pleasure enough in itself, 
why, every time I complete seventeen essays. Doubleday 
& Company, Inc., puts them into a book and publishes 
them. Twelve books of my F & SF essays have been pub- 
lished by now, containing a total of 204 essays. A thir- 
teenth, of course, is on its way. 

Few books, however, can be expected to sell indefinitely; 
at least, not well enough to be worth the investment of 
keeping them forever in print. The estimable gentlemen 
at Doubleday have therefore (with some reluctance, for 
they are fond of me and know how my lower lip tends to 
tremble on these occasions) allowed the first five of my 
books of essays to go out of print. 

Out of hardback print, I hasten to say. All five of the 
books are flourishing in paperback, so that they are still 
available to the public. Nevertheless, there is a cachet 


xi 



INTRODUCTION 


about the hardback that I am reluctant to lose. It is the 
hardbacks that supply the libraries; and for those who 
really want a permanent addition to their large personal 
collections of Asimov books * there is nothing like a 
hardback. 

My first impulse, then, was to ask the kind people at 
Doubleday to put the books back into print and gamble 
on a kind of second wind. This is done periodically in 
the case of my science fiction books, with success even 
when paperback editions are simultaneously available. But 
I could see that with my essays the case was different. My 
science fiction is ever fresh, but science essays do tend to 
get out of date, for the advance of science is inexorable. 

And then I got to thinking. . . . 

I deliberately range widely over the various sciences 
both to satisfy my own restless interests and to give each 
member of my heterogeneous audience a chance to satisfy 
his or her own particular taste now and then. The result 
is that each collection of essays has some on astronomy, 
some on chemistry, some on physics, some on biology, 
some on mathematics, and so on. 

But what about the reader who is interested in science 
but is particularly interested in one particular branch? He 
has to read through all the articles in the book to find 
three or four that may be just up his alley. 

Why not, then, go through the five out-of-print books, 
cull the articles on a particular branch of science, and 
put them together in a more specialized volume. 

Doubleday agreed and I made up a collection of as- 
tronomical articles which appeared as Asimov on Astron- 
omy. This introduction appeared at the start, explaining 
to the readers just what I had done and why 1 had done it, 
so that everything was open and on the table. It was an 
experimental project, of course, and might have done 
very poorly. It did not; it did very well. The people at 
Doubleday rejoiced and I got to work at once (grinning) 
and put together Asimov on Chemistry and Asimov on 
Physics, each with the same introduction, as in the first 
one of this series, so that the reader will continue to be 
warned of what is going on. 


Well , start one ! 


Introduction 


xiii 

These, too, did well, and now I am preparing a fourth 
and last * of these volumes, Asimov on Numbers , which 
contains one article from View from a Height, seven ar- 
ticles from Adding a Dimension , four articles from Of 
Time and Space and Other Things , and five articles from 
Barth to Heaven. The articles are arranged, not chrono- 
logically, but conceptually. 

Aside from grouping the articles into a more homogen- 
eous mass in an orderly arrangement, what more have I 
done? Well, the articles are anywhere from nine to sixteen 
years old and their age shows here and there. I feel rather 
pleased that the advance of science has not knocked out 
a single one of the articles here included, or even seriously 
dented any, but minor changes must be made, and I have 
made them. 

In doing this, I have tried not to revise the articles 
themselves since that would deprive you of the fun of 
seeing me eat my words now and then, or, anyway, chew 
them a little. So I have made changes by adding footnotes 
here and there where something I said needed modifica- 
tion or where I was forced to make a change to avoid 
presenting misinformation in the course of the article. 
Where it was necessary to make a number of small changes 
in the statistics and doing it in footnotes would be unbear- 
ably clumsy, I did revise the article, but in that case I 
warn the reader I am doing so. 

In addition to that, my good friends at Doubleday have 
decided to prepare these books on the individual branches 
of science in a consistent and more elaborate format than 
they have use for my ordinary essay collections and have 
added illustrations to which I have written captions that 
give information above and beyond what is in the essays 
themselves. 

Finally, since the subject matter is so much more 
homogeneous than in my ordinary grab-bag essay collec- 
tions, I have prepared an index (though I must admit that 
this will be less useful in this case than in the first three 
books of the series). 

* There are , of course , seventeen essays in the five out-of-print books 
that have appeared in none of the four collections, but they are a miscel- 
laneous lot . Perhaps when more of my essay books go out of print , I can 
combine some of them with additional articles from the later books to 
add to the subject-centered collections . 



xiv 


INTRODUCTION 


So, although the individual essays are old, I hope you 
find the book new and enjoyable just the same. And at 
least I have explained, in all honesty, exactly what I have 
done and why. The rest is up to you. 

Isaac Asimov 

New York, November 1975 


mm 

HM 


Part I 


NUMBERS AND 
COUNTING 




1 NOTHING 
COUNTS 


Roman numerals seem, even after five centuries of 
obsolescence, to exert a peculiar fascination over the in- 
quiring mind. 

It is my theory that the reason for this is that Roman 
numerals appeal to the ego. When one passes a corner- 
stone which says: “Erected MCMXVIII,” it gives one a 
sensation of power to say, “Ah, yes, nineteen eighteen” 
to one’s self. Whatever the reason, they are worth further 
discussion. 

The notion of number and of counting, as well as the 
names of the smaller and more-often-used numbers, dates 
back to prehistoric times and I don’t believe that there is 
a tribe of human beings on Earth today, however primi- 
tive, that does not have some notion of number. 

With the invention of writing (a step which marks the 
boundary line between “prehistoric” and “historic”), the 
next step had to be taken — numbers had to be written. 
One can, of course, easily devise written symbols for the 
words that represent particular numbers, as easily as for 
any other word. In English we can write the number of 
fingers on one hand as “five” and the number of digits on 
all four limbs as “twenty.” 

Early in the game, however, the kings' tax-collectors, 

3 



4 


NUMBERS AND COUNTING 


chroniclers, and scribes saw that numbers had the pecu- 
liarity of being ordered. There was one set way of count- 
ing numbers and any number could be defined by counting 
up to it. Therefore why not make marks which need be 
counted up to the proper number. 

Thus, if we let “one” be represented as / and “two” as ", 
and “three” as we can then work out the number indi- 
cated by a given symbol without trouble. You can see, for 
instance, that the symbol *********************** stands for “twenty- 
three.” What’s more, such a symbol is universal. What- 
ever language you count in, the symbol stands for the 
number “twenty-three” in whatever sound your particular 
language uses to represent it. 

It gets hard to read too many marks in an unbroken 
row, so it is only natural to break it up into smaller groups. 
If we are used to counting on the fingers of one hand, it 
seems natural to break up the marks into groups of five. 
“Twenty-three” then becomes If we are 

more sophisticated and use both hands in counting, we 
would write it ********** wtmw w we g Q barefoot and use 
our toes, too, we might break numbers into twenties. 

All three methods of breaking up number symbols into 
more easily handled groups have left their mark on the 
various number systems of mankind, but the favorite was 
division into ten. Twenty symbols in one group are, on 
the whole, too many for easy grasping, while five symbols 
in one group produce too many groups as numbers grow 
larger. Division into ten is the happy compromise. 

It seems a natural thought to go on to indicate groups 
of ten by a separate mark. There is no reason to insist on 
writing out a group of ten as ********** every time, when a 
separate mark, let us say can be used for the purpose. 
In that case “twenty-three” could be written as 

Once you’ve started this way, the next steps are clear. 
By the time you have ten groups of ten (a hundred), you 
can introduce another symbol, for instance -f. Ten hun- 
dreds, or a thousand, can become — and so on. In that 
case, the number “four thousand six hundred seventy-five” 
can be written = = = = + + + + + + 

To make such a set of symbols more easily graspable, 
we can take advantage of the ability of the eye to form a 
pattern. (You know how you can tell the numbers dis- 
played by a pack of cards or a pair of dice by the pattern 


Nothing Counts 5 

itself.) We could therefore write “four thousand six hun- 
dred seventy-five” as 


-h 4H 9 . 

And, as a matter of fact, the ancient Babylonians used 
just this system of writing numbers, but they used cunei- 
form wedges to express it. 

The Greeks, in the earlier stages of their development, 
used a system similar to that of the Babylonians, but in 
later times an alternate method grew popular. They made 
use of another order system— that of the letters of the 
alphabet. 

It is natural to correlate the alphabet and the number 
system. We are taught both about the same time in child- 
hood, and the two ordered systems of objects naturally 
tend to match up. The series “ay, bee, see, dee . . comes 
as glibly as “one, two, three, four . . and there is no 
difficulty in substituting one for the other. 

If we use undifferentiated symbols such as for 

“seven,” all the components of the symbol are identical 
and all must be included without exception if the symbol 
is to mean “seven” and nothing else. On the other hand, 
if “ABCDEFG” stands for “seven” (count the letters and 
see) then, since each symbol is different, only the last need 
to be written. You can’t confuse the fact that G is the 
seventh letter of the alphabet and therefore stands for 
“seven.” In this way, a one-component symbol does the 
work of a seven-component symbol. Furthermore, """ 
(six) looks very much like (seven); whereas F (six) 
looks nothing at all like G (seven). 

The Greeks used their own alphabet, of course, but let’s 
use our alphabet here for the complete demonstration: 
A=one, B— two, C— three, D^four, E— five, F— six, G— 
seven, H— eight, I— nine, and J=ten. 

We could let the letter K go on to equal “eleven,” but 
at that rate our alphabet will only help us up through 
“twenty-six.” The Greeks had a better system. The Baby- 
lonian notion of groups of ten had left its mark. If J=ten, 
than J equals not only ten objects but also one group of 



6 


NUMBERS AND COUNTING 


Nothing Counts 


7 


tens? Why not, then, continue the next letters as num- 
bering groups of tens? 

In other words J=ten, twenty, L~ thirty, M=forty, 
N= fifty, 0= sixty, P- seventy, eighty, R- ninety. Then 

we can go on to number groups of hundreds: S=one 
hundred, T=two hundred, U= three hundred, V=four 
hundred, W— five hundred, X=six hundred, Y— seven 
hundred, Z— eight hundred. It would be convenient to go 
on to nine hundred, but we have run out of letters. How- 
ever, in old-fashioned alphabets the ampersand {&) was 
sometimes placed at the end of the alphabet, so we can 
say that &=nine hundred. 

The first nine letters, in other words, represent the 
units from one to nine, the second nine letters represent 
the tens groups from one to nine, the third nine letters 
represent the hundreds groups from one to nine. (The 
Greek alphabet, in classic times, had only twenty-four let- 
ters where twenty-seven are needed, so the Greek made 
use of three archaic letters to fill out the list.) 

This system possesses its advantages and disadvantages 
over the Babylonian system. One advantage is that any 
number under a thousand can be given in three symbols. 
For instance, by the system I have just set up with our 
alphabet, six hundred seventy-five is XPE, while eight 
hundred sixteen is ZJF. 

One disadvantage of the Greek system, however, is that 
the significance of twenty-seven different symbols must be 
carefully memorized for the use of numbers to a thousand, 
where as in the Babylonian system only three different 
symbols must be memorized. 

Furthermore, the Greek system comes to a natural end 
when the letters of the alphabet are used up. Nine hun- 
dred ninety-nine (&RI) is the largest number that can be 
written without introducing special markings to indicate 
that a particular symbol indicates groups of thousands, 
tens of thousands, and so on. I will get back to this later. 

A rather subtle disadvantage of the Greek system was 
that the same symbols were used for numbers and words 
so that the mind could be easily distracted. For instance, 
the Jews of Graeco-Roman times adopted the Greek sys- 
tem of representing numbers but, of course, used the 
Hebrew alphabet — and promptly ran into difficulty. The 
number “fifteen 1 ’ would naturally be written as “ten-five.” 


In the Hebrew alphabet, however, “ten-five” represents a 
short version of the ineffable name of the Lord, and the 
Jews, uneasy at the sacrilege, allowed “fifteen” to be repre- 
sented as “nine-six” instead. 

Worse yet, words in the Greek-Hebrew system look like 
numbers. For instance, to use our own alphabet, WRA 
is “five hundred ninety- one.” In the alphabet system it 
doesn't usually matter in which order we place the sym- 
bols though, as we shall see, this came to be untrue for 
the Roman numerals, which are also alphabetic, and WAR 
also means “five hundred ninety-one.” (After all, we can 
say “five hundred one-and-ninety” if we wish.) Conse- 
quently, it is easy to believe that there is something war- 
like, martial, and of ominous import in the number “five 
hundred ninety-one,” 

The Jews, poring over every syllable of the Bible in their 
effort to copy the word of the Lord with the exactness that 
reverence required, saw numbers in all the words, and in 
New Testament times a whole system of mysticism arose 
over the numerical inter-relationships within the Bible. 
This was the nearest the Jews came to mathematics, and 
they called this numbering of words gematria, which is a 
distortion of the Greek geometria r We now call it “nu- 
merology.” 

Some poor souls, even today, assign numbers to the 
different letters and decide which names are lucky and 
which unlucky, and which boy should marry which girl 
and so on. It is one of the more laughable pseudo-sciences. 

In one case, a piece of gematria had repercussions in 
later history. This bit of gematria is to be found in “The 
Revelation of St. John the Divine,” the last book of the New 
Testament — a book which is written in a mystical fashion 
that defies literal understanding. The reason for the lack 
of clarity seems quite clear to me. The author of Revela- 
tion was denouncing the Roman government and was lay- 
ing himself open to a charge of treason and to subsequent 
crucifixion if he made his words too clear. Consequently, 
he made an effort to write in such a way as to be perfectly 
clear to his “in-group” audience, while remaining com- 
pletely meaningless to the Roman authorities. 

In the thirteenth chapter he speaks of beasts of diaboli- 
cal powers, and in the eighteenth verse he says, “Here is 
wisdom. Let him that hath understanding count the num- 



8 


NUMBERS AND COUNTING 


Nothing Counts 


9 


her of the beast: for it is the number of a man; and his 
number is Six hundred three-score and six.” 

Clearly, this is designed not to give the pseudo-science 
of gematria holy sanction, but merely to serve as a guide 
to the actual person meant by the obscure imagery of the 
chapter. Revelation, as nearly as is known, was written 
only a few decades after the first great persecution of 
Christians under Nero. If Nero's name (“Neron Caesar”) 
is written in Hebrew characters the sum of the numbers 
represented by the individual letters does indeed come out 
to be six hundred sixty-six, “the number of the beast,” 

Of course, other interpretations are possible. In fact, if 
Revelation is taken as having significance for all time as 
well as for the particular time in which it was written, it 
may also refer to some anti-Christ of the future. For this 
reason, generation after generation, people have made at- 
tempts to show that, by the appropriate jugglings of the 
spelling of a name in an appropriate language, and by the 
appropriate assignment of numbers to letters, some partic- 
ular personal enemy could be made to possess the number 
of the beast. 

If the Christians could apply it to Nero, the Jews them- 
selves might easily have applied it in the next century to 
Hadrian, if they had wished. Five centuries later it could 
be (and was) applied to Mohammed. At the time of the 
Reformation, Catholics calculated Martin Luther's name 
and found it to be the number of the beast, and Protestants 
returned the compliment by making the same discovery 
in the case of several popes. 

Later still, when religious rivalries were replaced by 
nationalistic ones, Napoleon Bonaparte and William II 
were appropriately worked out. What's more, a few min- 
utes' work with my own system of alphabet-numbers shows 
me that “Herr Adollf Hitler” has the number of the beast, 
(I need that extra “1” to make it work.) 

The Roman system of number symbols had similarities 
to both the Greek and Babylonian systems. Like the 
Greeks, the Romans used letters of the alphabet. However, 
they did not use them in order, but used just a few letters 
which they repeated as often as necessary — as in the 
Babylonian system. Unlike the Babylonians, the Romans 
did not invent a new symbol for every tenfold increase of 
number, but (more primitively) used new symbols for five- 
fold increases as well. 


Thus, to begin with, the symbol for “one” is I, and 
“two,” “three ” and “four,” can be written II, III, and IIII. 

The symbol for five, then, is not IIIII, but V. People 
have amused themselves no end trying to work out the 
reasons for the particular letters chosen as symbols, but 
there are no explanations that are universally accepted. 
However, it is pleasant to think that I represents the up- 
held finger and that V might symbolize the hand itself 
with all five fingers — one branch of the V would be the 
outheld thumb, the other, the remaining fingers. For “six,” 
“seven,” “eight,” and “nine,” we would then have VI, 
VII, VIII, and VIIII, 

For “ten” we would then have X, which (some people 
think) represents both hands held wrist to wrist. “Twenty- 
three” would be XXIII, “forty-eight” would be XXXXVIII, 
and so on. 

The symbol for “fifty” is L, for “one hundred” is C, 
for “five hundred” is D, and for “one thousand” is M. 
The C and M are easy to understand, for C is the first 
leter of centum (meaning “one hundred”) and M is the 
first letter of mille (one thousand). 

For that very reason, however, those symbols are sus- 
picious. As initials they may have come to oust the origi- 
nal less-meaningful symbols for those numbers. For 
instance, an alternative symbol for “thousand” looks some- 
thing like this (I). Half of a thousand or “five hundred” 
is the right half of the symbol, or I) and this may have 
been converted into D. As for the L which stands for 
“fifty,” I don't know why it is used. 

Now, then, we can write nineteen sixty- four, in Roman 
numerals, as follows: MDCCCCLXIIII. 

One advantage of writing numbers according to this 
system is that it doesn't matter in which order the num- 
bers are written. If I decided to write nineteen sixtv-four 
as CDCLIIMXCICI, it would still represent nineteen 
sixty-four if I add up the number values of each letter. 
However, it is not likely that anyone would ever scramble 
the letters in this fashion. If the letters were written in 
strict order of decreasing value, as I did the first time, it 
would then be much simpler to add the values of the let- 
ters. And, in fact, this order of decreasing value is (except 
for special cases) always used. 

Once the order of writing the letters in Roman numerals 
is made an established convention, one can make use of 



10 


NUMBERS AND COUNTING 


ROMAN NUMERALS 

This is a horoscope cast for Albrecht von Wallenstein, an 
Imperial general during the Thirty Years' War , by the 
great astronomer Johann Kepler . ( Kepler cast horoscopes 
in order to make a living, just as a modern actor , even a 
good one, might do commercials on the side.) 

Even though the numerals used are mostly Arabic , the 
twelve signs of the zodiac are numbered in Roman style 
for their greater effect. Roman numerals carried a cachet 
of stateliness for centuries after they were seen to be use- 
less in computation. 

Although our own familiar system is based on 10 and 
powers of 10 , the Roman numerals are based on both 5 
and 10 with special symbols for 1, 5, 10, 50, 100, 500, and 
1 } 000. Obviously f this arises because we have five fingers 
on each hand and ten fingers altogether. 

In a barefoot society it doesn't take much of an intel- 
lectual jump to decide to base a number system on 20. 
The ■ Mayans of Central America counted by both tens 
and twenties and had special symbols for 20, for 400 
(20% 8,000 (20% 160,000 (20*) and so on. 

Although there is no formal vigesimal (by twenties) 
number system in Western tradition , we still count by 
“scores” and say " four score and seven” (four twenties and 
seven) when we mean 87 . Counting by twenties is so 
common, in fact, that we speak of keeping score and ask 
et What's the score?" in connection with a ball game . 

Duodecimal systems are also used, in words anyway, if 
not in symbols, because 12 can be divided evenly by 2, 
3, 4, and 6. We speak of dozens, therefore, and of grosses , 
where a gross is a dozen dozen, or 144 . For that matter , 
the ancient Sumerians used a 60~based system and we 
still have 60 seconds to the minute and 60 minutes to the 
hour . 


Nothing Counts 


11 


^ovofcophm jgefidiet butd) 
loarmem Kepplerum 

i 6 o 8 . 



m. iv. v. 


The Granger Collection 




12 


NUMBERS AND COUNTING 


Nothing Counts 


13 


deviations from that set order if it will help simplify mat- 
ters, For instance, suppose we decide that when a symbol 
of smaller value follows one of larger value, the two are 
added; while if the symbol of smaller value precedes one 
of larger value, the first is subtracted from the second. 
Thus VI is "five" plus "one" or “six," while IV is “five'’ 
minus “one" or “four.” (One might even say that IIV is 
“three,” but it is conventional to subtract no more than 
one symbol,) In the same way LX is “sixty” while XL is 
“forty”; CX is “one hundred ten,” while XC is “ninety”; 
MC is “one thousand one hundred,” while CM is “nine 
hundred.” 

The value of this “subtractive principle” is that two 
symbols can do the wort of five. Why write VIIII if you 
can write IX; or DCCCC if you can write CM? The year 
nineteen sixty-four, instead of being written MDCCCC- 
LXIIII (twelve symbols), can be written MCMLXIV 
(seven symbols). On the other hand, once you make the 
order of writing letters significant, you can no longer 
scramble them even if you wanted to. For instance, if 
MCMLXIV is scrambled to MMCLXVI it becomes “two 
thousand one hundred sixty-six.” 

The subtractive principle was used on and off in ancient 
times but was not regularly adopted until the Middle Ages. 
One interesting theory for the delay involves the simplest 
use of the principle — that of IV (“four”). These are the 
first letters of IV PITER the chief of the Roman gods, 
and the Romans may have had a delicacy about writing 
even the beginning of the name. Even today, on clockfaces 
bearing Roman numerals, “four” is represented as IIII and 
never as IV. This is not because the clockface does not 
accept the subtractive principle, for “nine” is represented 
as IX and never as VIIII. 

With the symbols already given, we can go up to the 
number “four thousand nine hundred ninety-nine” in Ro- 
man numerals. This would be MMMMDCCCCLXXXX- 
VIIII or, if the subtractive principle is used, MMMMCM- 
XCIX. You might suppose that “five thousand” (the next 
number) could be written MMMMM, but this is not quite 
right. Strictly speaking, the Roman system never requires 
a symbol to be repeated more than four times. A new sym- 
bol is always invented to prevent that: IIIII— V; XXXXX 
=L; and CCCCC— D. Well, then, what is MMMMM? 

No letter was decided upon for “five thousand.” In an- 


cient times there was little need in ordinary life for num- 
bers that high. And if scholars and tax collectors had oc- 
casion for large numbers, their systems did not percolate 
down to the common man. 

One method of penetrating to “five thousand” _and be- 
yond is to use a bar to represent thousands. Thus, V would 
represent not “five” but “five thousand.” And sixty-seven 
thousand four hundred eighty-two would be LXVIICD- 
LXXXII. 

But another method of writing large numbers harks back 
to the primitive symbol (I) for “thousand.” By adding to 
the curved lines we can increase the number by ratios of 
ten. Thus “ten thousand” would be ((I)), and “one hun- 
dred thousand” would be (((I))), Then just as “five hun- 
dred” was I) or D, “five thousand” would be I)) and 
“fifty thousand” would be I))). 

Just as the Romans made special marks to indicate thou- 
sands, so did the Greeks. What’s more, the Greeks made 
special marks for ten thousands and for millions (or at 
least some of the Greek writers did). That the Romans 
didn’t carry this to the logical extreme is no surprise. The 
Romans prided themselves on being non-intellectual. That 
the Greeks missed it also, however, will never cease to 
astonish me. 

Suppose that instead of making special marks for large 
numbers only, one were to make special marks for every 
type of group from the units on. If we stick to the system 
I introduced at the start of the chapter — that is, the one 
in which ' stands for units, - for tens, for hundreds, 
and = for thousands — then we could get by with but one 
set of nine symbols. We could write every number with a 
little heading, marking off the type of groups — Then 

for “two thousand five hundred eighty-one” we could get 

= + - / 

by with only the letters from A to I and write it BEHA. 
What’s more for “five thousand five hundred fifty-five” 

= + * f 

we could write EEEE. There would be no confusion with 
all the E’s since the symbol above each E would indicate 
that one was a “five,” another a “fifty,” another a “five 
hundred,” and another a “five thousand.” By using addi- 
tional symbols for ten thousands, hundred thousands, mil- 
lions, and so on, any number, however large, could be 
written in this same fashion. 

Vet it is not surprising that this would not be popular. 



14 


NUMBERS AND COUNTING 


Nothing Counts 


15 


Even if a Greek had thought of it he would have been 
repelled by the necessity of writing those tiny symbols. In 
an age of hand-copying, additional symbols meant addi- 
tional labor and scribes would resent that furiously. 

Of course, one might easily decide that the symbols 
weren’t necessary. The groups, one could agree, could al- 
ways be written right to left in increasing values. The 
units would be at the right end, the tens next on the left, 
the hundreds next, and so on. In that case BEHA would 
be “two thousand five hundred eighty-one” and EEEE 
would be “five thousand five hundred fifty-five” even with- 
out the little symbols on top. 

Here, though, a difficulty would creep in. What if there 
were no groups of ten, or perhaps no units, in a particular 
number? Consider the number “ten” or the number “one 
hundred and one.” The former is made up of one group of 
ten and no units, while the latter is made up of one group 
of hundreds, no groups of tens, and one unit. Using sym- 
bols over the columns, the numbers could be written A 
+ • / 

and A A, but now you would not dare leave out the little 
symbols. If you did, how could you differentiate A mean- 
ing "ten” from A meaning “one” or A A meaning “one 
hundred and one” from AA meaning “eleven” or AA 
meaning “one hundred and ten”? 

You might try to leave a gap so as to indicate “one 
hundred and one” by A A. But then, in an age of hand- 
copying, how quickly would that become AA, or, for that 
matter, how quickly might AA become A A? Then, too, 
how would you indicate a gap at the end of a symbol? No, 
even if the Greeks throught of this system, they must ob- 
viously have come to the conclusion that the existence of 
gaps in numbers made this attempted simplification im- 
practical. They decided it was safer to let I stand for “ten” 
and SA for “one hundred and one” and to Hades with 
little symbols. 

What no Greek ever thought of — not even Archimedes 
himself — was that it wasn’t absolutely necessary to work 
with gaps. One could fill the gap with a symbol by letting 
one stand for nothing — for “no groups.” Suppose we use 
$ as such a symbol. Then, if “one hundred and one” is 
made up of one group of hundreds, no groups of tens, and 
one unit, it can be written ASA. If we do that sort of 
thing, all gaps are eliminated and we don’t need the little 


symbols on top. “One” becomes A, “ten” becomes A$, 
“one hundred” becomes A$$, “one hundred and one” be- 
comes A$A, “one hundred and ten” becomes AA$, and 
so on. Any number, however large, can be written with the 
use of exactly nine letters plus a symbol for nothing. 

Surely this is the simplest thing in the world — after you 
think of it. 

Yet it took men about five thousand years, counting 
from the beginning of number symbols, to think of a sym- 
bol for nothing. The man who succeeded (one of the most 
creative and original thinkers in history) is unknown. We 
know only that he was some Hindu who lived no later 
than the ninth century. 

The Hindus called the symbol sunya> meaning “empty.” 
This symbol for nothing was picked up by the Arabs, who 
termed it sifr , which in their language meant “empty,” 
This has been distorted into our own words “cipher” and, 
by way of zefirum , into “zero.” 

Very slowly, the new system of numerals (called “Ara- 
bic numerals” because the Europeans learned of them from 
the Arabs) reached the West and replaced the Roman 
system. 

Because the Arabic numerals came from lands which 
did not use the Roman alphabet, the shape of the numerals 
was nothing like the letters of the Roman alphabet and 
this was good, too. It removed word-number confusion 
and reduced gematria from the everyday occupation of 
anyone who could read, to a burdensome folly that only a 
few would wish to bother with. 

The Arabic numerals as now used by us are, of course, 
1, 2, 3, 4, 5, 6, 7, 8, 9, and the all-important 0. Such is 
our reliance on these numerals (which are internationally 
accepted) that we are not even aware of the extent to 
which we rely on them. For instance if this chapter has 
seemed vaguely queer to you, perhaps it was because I 
had deliberately refrained from using Arabic numerals all 
through. 

We all know the great simplicity Arabic numerals have 
lent to arithmetical computation. The unnecessary load 
they took off the human mind, all because of the presence 
of the zero, is simply incalculable. Nor has this fact gone 
unnoticed in the English language. The importance of the 
zero is reflected in the fact that when we work out an 
arithmetical computation we are (to use a term now 



16 


NUMBERS AND COUNTING 


ARABIC NUMERALS 

Quite apart from the greater ease of computation with 
Arabic numerals t as compared with any other system 
man has invent ed, there is the compactness of it. Imagine 
all the numerical information in the table given here 
translated into Roman numerals (or any other kind). It 
would become a bulky mass that only an expert could 
make any sense of. 

For instance , it is obvious just in the number of digits 
of the number that 12,000 is greater than 787. You can 
go down the final column of the illustration in one quick 
sweep of the eye and see at a glance that the greatest 
number of sales listed for any item in it is the 285,800 
for Montgomery Ward. It happens to be the only six-digit 
number in the column that starts with a numeral higher 
than 1 . You don't even have to read the other digits to 
know ifs highest . 

This cannot be done in any other number system , For 
instance , of the two numbers XVIII and XL, the two- 
symbol number is over twice as great as the five-symbol 
number . 

Of course t there are disadvantages to the Arabic nu- 
meral system , too . There is no redundancy in it. Every 
digit has one and only one value , and every place has one 
and only one value. Drop a single digit or interchange a 
digit and you are lost. For instance t there is redund- 
ancy in words. Leave a letter out of “redundancy” and 
you have “redundncy” and hardly anyone will fail to see 
the correct word. Or invert two letters and you have 
“ redundancy ” and people see the mistake and allow for it. 

On the other hand , change 2835 to 235 by dropping 
the 8, or to 2385 by inverting two digits, and there is no 
sign that any mistake has been made or any chance of 
retrieving the correct value. 

This illustration, by the way, shows the famous stock 
market crash of 1929. Midland Steel Products f preferred, 
dropped 60 points. Murray Corp. was at 20 from a year’s 
high of lOQYe. Oh, boy. 


STO CttfeX CHANCE 


Nothing Counts 


17 


ii§i V ij *•**? j si j».*l 


if sg , 4 T 8a »5r *a' OT ¥ , f 

j * ft a- 5 ™ -^yEas&.-af-S" "i. A ' _T i £5F -a ■■ rH i*~~ ' ii' jiul v ' S if?.. _ 


: 1 , a e 

If' 


MW 




'■ ^ .it * ' lg 3^f,% ^ ^ ^ 




it 


nfnf atts. . L . * w-a-s* 






The Granger Collection 



18 


NUMBERS AND COUNTING 


slightly old-fashioned) ‘'ciphering.” And when we work 
out some code, we are “deciphering” it. 

So if you look once more at the title of this chapter, 
you will see that I am not being cynical. I mean it literally. 
Nothing counts! The symbol for nothing makes all the 
difference in the world. 



ONE, TEN, 
BUCKLE 
MY SHOE 


I have always been taken aback a little at my in- 
ability to solve mathematical conundrums since (in my 
secret heart of hearts) I feel this to be out of character 
for me. To be sure, numerous dear friends have offered 
the explanation that, deep within me, there rests an artfully 
concealed vein of stupidity, but this theory has somehow 
never commended itself to me. 

Unfortunately, I have no alternate explanation to sug- 
gest. 

You can well imagine, then, that when I come across a 
puzzle to which I can find the answer, my heart fairly 
sings. This happened to me once when I was quite young 
and I have never forgotten it. Let me explain it to you in 
some detail because it will get me somewhere I want to go. 

The problem, in essence, is this. You are offered any 
number of unit weights: one-gram, two-gram, three-gram, 
four-gram, and so on. Out of these you may choose a 
sufficient number so that by adding them together in the 
proper manner, you may be able to weigh out any integral 
number of grams from one to a thousand. Well, then, how 
can you choose the weights in such a way as to end with 
the fewest possible number that will turn the trick? 

I reasoned this way — 

I must start with a 1-gram weight, because only by using 
19 



20 


NUMBERS AND COUNTING 


One, Ten, Buckle My Shoe 


21 


it can I weigh out one gram. Now if I take a second 1- 
gram weight, I can weigh out two grams by using both 

1- gram weights. However, I can economize by taking a 

2- gram weight instead of a second 1-gram weight, for then 
not only can I weigh out two grams with it, but I can also 
weigh out three grams, by using the 2-gram plus the 1- 
gram. 

What’s next? A 3-gram weight perhaps? That would be 
wasteful, because three grams can already be weighed out 
by the 2-gram plus the 1-gram. So I went up a step and 
chose a 4-gram weight. That gave me not only the possi- 
bility of weighing four grams, but also five grams (4-gram 
plus 1-gram), six grams (4-gram plus 2-gram), and seven 
grams (4-gram plus 2-gram plus 1-gram). 

By then I was beginning to see a pattern. If seven grams 
was the most I could reach, I would take an 8 -gram weight 
as my next choice and that would carry me through each 
intergral weight to fifteen grams (8-grams plus 4-grams 
plus 2 -gram plus 1-gram). The next weight would be a 16- 
gram one, and it was clear to me that in order to weigh 
out any number of grams one bad to take a series of 
weights (beginning with the 1-gram) each one of which 
was double the next smaller. 

That meant that I could weigh out any number of grams 
from one to a thousand by means of ten and only ten 
weights: a 1-gram, 2-gram, 4-gram, 8-gram, 16-gram^ 32- 
gram, 64-gram, 128-gram, 256-gram, and 512-gram. In 
fact, these weights would carry me up to 1023 grams. 

Now we can forget weights and work with numbers 
only. Using the numbers 1, 2, 4, 8, 16, 32, 64, 128, 256, 
and 512, and those only, you can express any other num- 
ber up to and including 1023 by adding two or more of 
them. For instance the number 100 can be expressed as 
64 plus 32 plus 4. The number 729 can be expressed as 
512 plus 128 plus 64 plus 16 plus 8 plus 1. And, of 
course, 1023 can be expressed as the sum of all ten num- 
bers. 

If you add to this list of numbers 1024, then you can 
continue forming numbers up to 2047; and if you next add 
2048, you can continue forming numbers up to 4095; and 
if you next — 

Weil, if you start with 1 and continue doubling indefi- 
nitely, you will have a series of numbers which, by appro- 


priate addition, can be used to express any finite number 
at all. 

So far, so good; but our interesting series of numbers — 

1 2, 4, 8, 16, 32, 64, . . . — seems a little miscellaneous. 
Surely there must be a neater way of expressing it. And 
there is. 

Let’s forget 1 for a minute and tackle 2. If we do that, 
we can begin with the momentous statement that 2 is 2. 
(Any argument?) Going to the next number, we can say 
that 4 is 2 times 2. Then 8 is 2 times 2 times 2; 16 is 2 
times 2 times 2 times 2; 32 is . . . But you get the idea. 

So we can set up the series (continuing to ignore 1) as 
2, 2 times 2, 2 times 2 times 2, 2 times 2 times 2 times 2, 
and so on. There is a kind of pleasing uniformity and 
regularity about this but all those 2 times 2 times Ts 
create spots before the eyes. Therefore, instead of writing 
out all the 2’s, it would be convenient to note how many 
2’s are being multiplied together by using an exponential 
method. 

Thus, if 4 is equal to 2 times 2, we will call it 2- (two 
to the second power, or two squared) . Again if 8 is 2 times 

2 times 2, we can take note of the three 2’s multiplied to- 
gether by writing 8 as 2 3 (two to the third power, or two 
cubed). Following that line of attack we would have 16 
as 2 4 (two to the fourth power), 32 as 2 5 (two to the 
fifth power), and so on. As for 2 itself, only one 2 is in- 
volved and we can call it 2 1 (two to the first power). 

One more thing. We can decide to let 2° (two to the 
zero power) be equal to 1. (In fact, it is convenient to let 
any number to the zero power be equal to 1 . Thus, 3° 
equals 1, and so does 17° and 1,965,21 l fl . For the moment, 
however, we are interested only in 2° and we are letting 
that equal 1.) 

Well, then, instead of having the series 1, 2, 4, 8, 16, 
32, 64, . . . , we can have 2°, 2\ 2\ 2\ 2 4 , 2 n , 2 6 . . . . It's 
the same series as far as the value of the individual mem- 
bers are concerned, but the second way of writing it is 
prettier somehow and, as we shall see, more useful. 

We can express any number in terms of these powers 
of 2. I said earlier that 100 could be expressed as 64 plus 
32 plus 4. This means it can be expressed as 2 6 plus 2 5 
plus 2 2 . In the same way, if 729 is equal to 512 plus 12° 



22 


NUMBERS AND COUNTING 


One , Ten, Buckle My Shoe 


23 


plus 64 plus 16 plus 8 plus 1, then it can also be expressed 
as 2 3 plus 2 7 plus 2® plus 2 4 plus 2 3 plus 2°. And of course, 
1023 is 2° plus 2 s plus 2 7 plus 2 e plus 2 5 plus 2 4 plus 2 3 
plus 2 2 plus 2 1 plus 2°. 

But let’s be systematic about this* We are using ten dif- 
ferent powers of 2 to express any number below 1024, so 
let’s mention all of them as a matter of course. If we don’t 
want to use a certain power in the addition that is re- 
quired to express a particular number, then we need merely 
multiple it by 0* If we want to use it, we multiply it by 1, 
Those are the only alternatives; we either use a certain 
power, or we don’t use it; we either multiply it by 1 or 
by 0* 

Using a dot to signify multiplication, we can say that 
1023 is: 1'2 9 plus 1'2* plus 1'2 7 plus 1‘2 6 plus 1‘2 3 plus 1’2 4 
used* In expressing 729, however, we would have: 1*2° 
plus 1*2 3 plus 1’2 2 plus L2 1 plus T2 & . All the powers are 
used. In expressing 729, however, we would have: 1*2 Q 
plus 0‘2 8 plus 1’2 7 plus 1'2 6 plus 0 2 5 plus T2 4 plus V2 3 
plus 0*2 2 plus 0*2* plus T2°. And again, in expressing 100, 
we can write: 0*2 G plus 0'2 S plus 0*2 7 plus 1‘2 G plus 1*2 5 
plus 0*2 4 plus 0'2 3 plus V2 2 plus 0 2 1 plus 0*2°. 

But why bother, you might ask, to include those powers 
you don’t use? You write them out and then wipe them 
out by multiplying them by zero. The point is, however, 
that if you systematically write them all out, without" ex- 
ception, you can take it for granted that they are there and 
omit them altogether, keeping only the l’s and the 0’s. 

Thus, we can write 1023 as 1111111111; we can write 
729 as 101 1011 001; and we can write 100 as 0001100100. 

In fact, we can be systematic about this and, remember- 
ing the order of the powers, we can use the ten powers to 
express all the numbers up to 1023 this way: 

0000000001 equals 1 
0000000010 equals 2 
0000000011 equals 3 
0000000100 equals 4 
0000000101 equals 5 
0000000110 equals 6 
0000000111 equals 7, all the way up to 

llllilllll equals 1023 . 


Of course, we don’t have to confine ourselves to ten 
powers of 2, we can have eleven powers, or fourteen, or 
fifty-three, or an infinite number. However, it would get 
wearisome writing down an infinite number of l’s and 0’s 
just to indicate whether each one of an infinite number of 
powers of 2 is used or is not used. So it is conventional to 
leave out all the high powers of 2 that are not used for a 
particular number and just begin with the highest power 
that is used and continue from there. In other words, leave 
out the unbroken line of zeroes at the left. In that case, 
the numbers can be represented as 

1 equals 1 

10 equals 2 

11 equals 3 

100 equals 4 

101 equals 5 

110 equals 6 

1 1 1 equals 7, and so on. 

Any number at all can be expressed by some combina- 
tion of l’s and 0’s in this fashion, and a few primitive 

tribes have actually used a number system like this. The 

first civilized mathematician to work it out systematically, 
however, was Gottfried Wilhelm Leibniz, about three cen- 
turies ago. He was amazed and gratified because he rea- 
soned that 1 , representing unity, was clearly a symbol for 
God, while 0 represented the nothingness which, aside 
from God, existed in the beginning. Therefore, if all num- 
bers can be represented merely by the use of 1 and 0, 
surely this is the same as saying that God created the uni- 
verse out of nothing. 

Despite this awesome symbolism, this business of l’s 
and 0’s made no impression whatsoever on practical men 
of affairs. It might be a fascinating mathematical curiosity, 
but no accountant is going to work with 1011011001 in- 
stead of 729. 

But then it suddenly turned out that this two-based sys- 
tem of numbers (also called the “binary system,” from the 
Latin word binarius , meaning “two at a time”) is ideal 
for electronic computers. 

After all, the two different digits, 1 and 0, can be 
matched in the computer by the two different positions 



24 


NUMBERS AND COUNTING 


One , Ten t Buckle My Shoe 


25 


GOTTFRIED WILHELM LEIBNIZ 

Leibniz was born in Leipzig, Saxony , on July 1, 1646 , 
and was an amazing child prodigy . He taught himself 
Latin at eight and Greek at fourteen. He obtained a 
degree in law in 1665 and 9 in addition , was a diplomat > 
philosopher , political writer, and an attempted reconciler 
of Catholics and Protestants. On occasion he acted as 
adviser to Peter the Great of Russia . In 1671 he was the 
first person to devise a mechanical device that would 
multiply and divide as well as add and subtract . 

Leibniz visited London in 1673 and thereafter began 
to work out that branch of mathematics called calculus, 
which he published in 1684 . Isaac Newton had worked 
out the calculus independently at about the same time, 
but Newton was rather small-minded in all things where 
his genius didn't apply and he accused Leibniz of plagiar- 
ism. There was a long battle between defenders of the 
two men, but actually Leibniz's development was superior, 
and Great Britain, by sticking stubbornly to Newton, fell 
behind in mathematics and stayed behind for a century 
and a half. 

In 1700 Leibniz induced King Frederick 1 of Prussia 
to found the Academy of Sciences in Berlin and served 
as it first president . However, he spent almost all his 
mature years in the service of the Electors of Hanover . 
In 1714 the then-elector succeeded to the throne of 
Great Britain as George l, and Leibniz war eager to go 
with him to London . 

Kings are not notorious for anything but self-centered- 
ness t however , ana George 1 had no need of Leibniz. 
Leibniz died in Hanover on November 14 , J716, neglected 
and forgotten, with only his secretary attending the 
funeral. 



The Granger Collection 


26 


NUMBERS AND COUNTING 


One , Ten , Buckle My Shoe 


27 


of a particular switch: “on 1 ’ and “off.” Let “on” represent 
1 and “off” represent 0. Then, if the machine contained 
ten switches, the number 1023 could be indicated as on- 
on-on-on-on-on-on-omon-on; the number 729 could be 
on-off-on-on-off-on-on-off~off-on; and the number 100 
could be off-off-off-on-on-off-off-on-off-off. 

By adding more switches we can express any number 
we want simply by this on-off combination. It may seem 
complicated to us, but it is simplicity itself to the computer 
In fact, no other conceivable system could be as simple — 
for the computer. 

However, since we are only human beings, the question 
is, can we handle the two-based system? For instance, can 
we convert back and forth between two-based numbers 
and ordinary numbers? If we are shown 110001 in the 
two-based system, what does it mean in ordinary numbers? 

Actually, this is not difficult. The two-based system uses 
powers of 2. starting at the extreme right with 2° and 
moving up a power at a time as we move leftward. So we 
can write 1 10001 with little numbers underneath to rep- 
resent the exponents, thus 110001, Only the exponents 

543210 

under the i’s are used, so 110001 represents plus 2 4 
plus 2 r; or 32 plus 16 plus 1. In other words, 110001 in 
the two- based system is 49 in ordinary numbers. 

Working the other way is even simpler. You can, if you 
wish, try' to fit the powers of 2 into an ordinary number 
by hit and miss, but you don’t have to. There is a routine 
you can use which always works and I will describe it 
(though, if you will forgive me, I will not bother to explain 
why it works). 

Suppose you wish to convert an ordinary number into 
the two-based system. You divide it by 2 and set the re- 
mainder to one side. (If the number is even, the remainder 
will be zero; if odd, it will be 1.) Working only with 
the whole-number portion of the quotient, you divide that 
by 2 again, and again set the remainder to one side and 
work only with the whole-number portion of the new 
quotient. When the whole-number portion of the quotient 
is reduced to 0 as a result of the repeated divisions by 2, 
you stop. The remainders, read backward, give the original 
number in the two-based system. 


If this sounds complicated, it can be made simple 
enough by use of an example. Let’s try 131 : 

131 divided by 2 is 65 with a remainder of 1 
65 divided by 2 is 32 with a remainder of 1 
32 divided by 2 is 16 with a remainder of 0 
1 6 divided by 2 is 8 with a remainder of 0 
8 divided by 2 is 4 with a remainder of 0 
4 divided by 2 is 2 with a remainder of 0 
2 divided by 2 is 1 with a remainder of 0 
1 divided by 2 is 0 with a remainder of 1 

In the two-based system, then, 131 is written 10000011. 

With a little practice anyone who knows fourth-grade 
arithmetic can switch back and forth between ordinary 
numbers and two-based numbers. 

The two-based system has the added value that it makes 
the ordinary operations of arithmetic childishly simple. In 
using ordinary numbers, we spend several years in the 
early grades memorizing the fact that 9 plus 5 is 14, that 
8 times 3 is 24, and so on. 

In two-based numbers, however, the only digits involved 
are 1 and 0, so there are only four possible sums of digits 
taken two at a time: 0 plus 0, 1 plus 0, 0 plus 1, and 1 
plus L The first three are just what one would expect in 
ordinary arithmetic: 

0 plus 0 equals 0 

1 plus 0 equals 1 

0 plus 1 equals 1 

The fourth sum involves a slight difference. In ordinary 
arithmetic 1 plus 1 is 2, but there is no digit like 2 in the 
two-based system. There 2 is represented as 10. Therefore: 

1 plus 1 equals 10 (put down 0 and carry 1) 

Imagine, then, how simple addition is in the two-based 
system. If you want to add 1001101 and 11001, the sum 
would look like this: 

1001101 

11001 

nooiTo 



28 


NUMBERS AND COUNTING 


One , Ten , Buckle My Shoe 


29 


You can follow this easily from the addition table I’ve 
just given you, and by converting to ordinary numbers 
(as you ought also to be able to do) you will see that the 
addition is equivalent to 77 plus 25 equals 102. 

It may seem to you that following the l’s and 0’s is dif- 
ficult indeed and that the ease of memorizing the rules of 
addition is more than made up for by the ease of losing 
track of the whole thing. This is true enough — for a 
human. In a computer* however, on-off switches are easily 
designed in such combinations as to make it possible for 
the on’s and off’s to follow the rules of addition in the 
two-based system. Computers don’t get confused and 
surges of electrons bouncing this way and that add num- 
bers by two-based addition in microseconds. 

Of course (to get back to humans) if you want to add 
more than two numbers, you can always, at worst* break 
them up into groups of two. If you want to add 110, 101, 
100, and 111, you can first add 110 and 101 to get 1011, 

then add 100 and 111 to get 1011, and finally add 1011 

and 1011 to get 10110, (The last addition involves adding 
1 plus 1 plus 1 as a result of carrying a 1 into a column 

which is already 1 plus 1. Well, 1 plus 1 is 10 and 10 plus 

1 is 11, so 1 plus 1 plus 1 is 11, put down 1 and carry 1.) 

Multiplication in the two-based system is even simpler. 
Again, there are only four possible combinations: 0 times 
0, 0 times 1, 1 times 0, and 1 times 1. Here, each multipli- 
cation in the two-based system is exactly as it would be in 
ordinary numbers, fn other words: 

0 times 0 is 0 

0 times 1 is 0 

1 times 0 is 0 

1 times 1 is 1 

To multiply 101 by 1101, we would have 

101 

1101 

101 

000 

101 

101 

1000001 


In ordinary numbers, this is equivalent to saying 5 times 

is 65. Again, the computer can be designed to manipu- 
late the on’s and off’s of its switches to match the require- 
ments of the two-based multiplication table — and to do it 
with blinding speed. 

It is possible to have a number system based on powers 
0 f 3, also (a three-based or “ternary’ 1 system). The series 
of number 3°, 3\ 3\ 3 3 , 3 4 , and so on (that is* 1, 3, 9, 
27, 81, and so on) can be used to express any finite num- 
ber provided you are allowed to use up to two of each 
member of the series. 

Thus 17 is 9 plus 3 plus 3 plus 1 plus 1 and 72 is 27 
plus 27 plus 9 plus 9. 

If you wanted to write the series of integers according 
to the three-based system, they would be: 1, 2, 10, 11, 12, 
20 , 21 , 22 , 100 , 101 , 102 , 110 , 111 , 112 , 120 , 121 , 122 , 
200, and so on. 

You could have a four-based number system based on 
powers of 4, with each power used up to three times; a 
five-based number system based on power of 5 with each 
power used up to four times; and so on. 

To convert an ordinary number into any one of these 
other systems, you need only use a device similar to the 
one I have demonstrated for conversion into the two- 
based system, you would repeatedly divide by 3 for the 
three-based system, by 4 for the four-based system, and 
so on. 

Thus, I have already converted the ordinary number 
131 into 11000001 by dividing 131 repeatedly by 2 and 
using the remainders. Suppose we divide 131 repeatedly 
by 3 instead and make use of the remainders: 

131 divided by 3 is 43 with a remainder of 2 
43 divided by 3 is 14 with a remainder of 1 
14 divided by 3 is 4 with a remainder of 2 
4 divided by 3 is 1 with a remainder of 1 
1 divided by 3 is 0 with a remainder of 1 

The number 131 in the tbree-based system, then, is made 
up of the remainders, working from the bottom up, and 
is 11212. 

In similar fashion we can work out what 131 is in the 
four-based system, the five-based system, and so on. Here 



30 NUMBERS AND COUNTING 

is a little table to give you the values of 131 up through 
the nine-based system: a 


two-based system 

11000001 

three-based system 

11212 

four-based system 

2003 

five-based system 

1011 

six-based system 

335 

seven-based system 

245 

eight-based system 

203 

nine-based system 

155 


You can check these by working through the powers. 
In the nine-based system, 155 is V9 2 plus 5'9 l plus 5*9°. 
Since 9 2 is 81, 9 1 is 9, and 9° is 1, we have 81 plus 45 
plus 5, or 131. In the six-based system, 335 is 3*6 2 plus 
3-6 1 plus 5*6°. Since 6 2 is 36, 6 1 is 6, and 6° is 1, we have 
108 plus 18 plus 5, or 131. In the four-based system, 
2003 is 2*4 3 plus 0*4 2 plus 0*4* plus 3*4°, and since 4 3 is 64, 

4 2 is 16, 4 l is 4, and 4° is 1, we have 128 plus 0 plus 0 

plus 3, or 131. 

The others you can work out for yourself if you choose. 

But is there any point to stopping at a nine-based sys- 
tem? Can there be a ten-based system? Well, suppose we 
write 131 in the ten-based system by dividing it through 
by tens: 

131 divided by 10 is 13 with a remainder of 1 
13 divided by 10 is 1 with a remainder of 3 
1 divided by 10 is 0 with a remainder of 1 

And therefore 131 in the ten-based system is 131. 

In other words, our ordinary numbers are simply the 
ten-based system, working on a series of powers of 10: 
10°, 10 1 , 10-, 10 3 , and so on. The number 131 is equal to 
1T0 2 plus 3-10 1 plus 1*30°. Since 10 2 is 100, 10 l is 10, 
and 10° is 1, this means we have 100 plus 30 plus 1, 131. 

There is nothing basic or fundamental about ordinary 
numbers then. They are based on the powers of 10 because 
we have ten fingers and counted on our fingers to begin 
with, but the powers of any other number will fulfill all the 
mathematical requirements. 

Thus we can go on to an eleven-based system and a 


One , Ten, Buckle My Shoe 31 

twelve-based system. Here, one difficulty arises. The num- 
ber of digits (counting zero) that is required for any 
system is equal to the number used as base. 

In the two-based system, we need two different digits, 
0 and 1. In the three-based system, we need three different 
digits, 0, 1, and 2. In the familiar ten-based system, we 
need, of course, ten different digits, 0, 1, 2, 3, 4, 5, 6, 7, 
8, and 9. 

It follows, then, that in the eleven-based system we will 
need eleven different digits and in the twelve-based system 
twelve different digits. Let’s write @ for the eleventh digit 
and # for the twelfth. In ordinary ten-based numbers, 
@ is 10 and # is 11. 

Thus, 131 in the eleven-based system is: 

131 divided by 11 is 11 with a remainder of 10 (@) 

1 1 divided by 11 is 1 with a remainder of 0 

1 divided by 1 1 is 0 with a remainder of 1 

so that 131 in the eleven-based system is 10@. 

And in the twelve-based system : 

131 divided by 12 is 10 with a remainder of 11 (#) 

10 divided by 12 is 0 with a remainder of 10 (@) 

so that 131 in the twelve-based system is 

And we can go up and up and up and have a 4,583- 
based system if we wanted (but with 4,583 different digits, 
counting the zero) , 

Now all the number systems may be valid, but which 
system is most convenient? As one goes to higher and 
higher bases, numbers become shorter and shorter. Though 
131 is 11000001 in the two-based system, it is 131 in the 
ten-based system and @# in the twelve-based system. It 
moves from eight digits to three digits to two digits. In 
fact, in a 131-based system (and higher) it would be 
down to a single digit. In a way, this represents increasing 
convenience. Who needs long numbers? 

However, the number of different digits used in con- 
structing numbers goes up with the base and this is an 
increasing inconvenience. Somewhere there is an inter- 
mediate base in which the number of different digits isn’t 



32 


NUMBERS AND COUNTING 


One, Ten , Buckle My Shoe 


33 


COMPUTERS 

Computers have a bad press these days . They are sup * 
posed to be soulless and dehumanizing . But what do 
people expect? They insist on populating the world by the 
billions . They insist on a government that spends hundreds 
of billions, (Sure they do. Every single person in the 
United States is for less government spending except where 
that hurts his way of life; and every cut damages the liveli- 
hood of millions so there are no cuts.) They insist on big 
business, big science , big armies , and everything else, and 
things have grown so complicated that none of it is pos- 
sible without computers. 

Sure , computers make humorous mistakes, but that's 
not the computer , It f s the human being who programmed 
it or operated it. If you can't make your check stubs 


balance , do you blame the number system or your own 
inability to add? (The number system? Well , then, / sup- 
pose you can blame computers , too.) 

Dehumanizing ? I suspect the same complaint was made 
by some proto-Sumerian architect who got sick and tired 
of the knotted ropes that the young apprentices were 
carrying about to measure the distances on the temple 
under construction. An architect should use his mind and 
eyes , he would say, and not depend on soulless mechanical 
contrivances . 

Actually , people who say nasty things about computers 
are only indulging in a pseudo-intellectual cheap-shot . 
There's no way of removing them from society without 
disaster, and if all the computers went on strike for 
twenty-four hours, you would experience what one means 
by a completely stalled nation. 

It is safe to complain about our modern technology 
while taking full advantage of it. Costs nothing. 





34 


NUMBERS AND COUNTING 


too high and the number of digits in the usual numbers 
we use isn’t too great. 

Naturally it would seem to us that the ten-based system 
is just right. Ten different digits to memorize doesn’t 
seem too high a price to pay for using only four digit 
combinations to make up any number under ten thousand. 

Yet the twelve-based system has been touted now and 
then. Four digit combinations in the twelve-based system 
will carry one up to a little over twenty thousand, but 
that seems scarcely sufficient recompense for the task of 
learning to manipulate two extra digits. (School children 
would have to learn such operations as @ plus 5 is 13 
and # times 4 is 38.) 

But here another point arises. When you deal with any 
number system, you tend to talk in round numbers: 10, 
100, 1000, and so on. Well, 10 in the ten-based system is 
evenly divisible by 2 and 5 and that is all. On the other 
hand, 10 in the twelve-based system (which is equivalent 
to 12 in the ten-based system) is evenly divisible by 2, 3, 
4, and 6. This means that a twelve-based system would be 
more adaptable to commercial transactions and, indeed, 
the twelve-based system is used every time things are sold 
in dozens (12’s) and grosses (144’s) for 12 is 10 and 144 
is 100 in the twelve-based system. 

In this age of computers, however, the attraction is 
toward a two-based system. And while a two-based system 
is an uncomfortable and unaesthetic melange of l’s and 
0’s, there is a compromise possible. 

A two-based system is closely related to an eight-based 
system, for 1000 on the two-based system is equal to 10 
on the eight-based system, or, if you’d rather, 2 3 equals 8 1 . 
We could therefore set up a correspondence as follows: 

TWO-BASED SYSTEM EIGHT-BASED SYSTEM 

000 0 

001 1 

010 2 

Oil 3 

100 4 

101 5 

110 6 

111 7 


One, Ten , Buckle My Shoe 


35 


This would take care of all the digits (including zero) 
in the eight-based system and all the three-digit combina- 
tions (including 000) in the two-based system. 

Therefore any two-based number could be broken up 
into groups of three digits (with zeros added to the left 
if necessary) and converted into an eight-based number 
by using the table Tve just given you. Thus, the two-based 
number 111001000010100110 could be broken up as 111,- 
001,000,010,100,110 and written as the eight-based num- 
ber, 710246. On the other hand, the eight-based number 
33574 can be written as the two-based number 0110111- 
01111100 almost as fast as one can write, once one learns 
the table. 

In other words, if we switched from a ten-based system 
to an eight-based system, there would be a much greater 
understanding between ourselves and our machines and 
who knows how much faster science would progress. 

Of course, such a switch isn’t practical, but just think — 
Suppose that, originally, primitive man had learned to 
count on his eight fingers only and had left out those two 
awkward and troublesome thumbs. 



Exclamation Point! 


37 


3 EXCLAMATION 

POINT! 


It is a sad thing to be unrequitedly in love, I can 
tell you. The truth is that I love mathematics and mathe- 
matics is completely indifferent to me. 

Oh, I can handle the elementary aspects of math all 
right but as soon as subtle insights are required, she goes 
in search of someone else. She’s not interested in me. 

I know this because every once in a while I get all in- 
volved with pencil and paper, on the track of some great 
mathematical discovery and so far T have obtained onty 
two kinds of results: 1) completely correct findings that 
are quite old, and 2) completely new findings that are 
quite wrong. 

For instance (as an example of the first class of results). 
I discovered, when I was very young, that the sums of 
successive odd numbers were successive squares. In other 
words: 1 = 1; 1+3=4; 1 +3 + 5=9; 1 + 3 + 5+7=16, and 
so on. Unfortunately, Pythagoras knew' this too in 500 
B.C., and I suspect that some Babylonian knew it in 1500 
B.C, 

An example of the second kind of result involves 
Fermat’s Last Theorem.* I was thinking about it a couple 


* Vm not going to discuss that here . Suffice it to say now that it is the 
most famous unsolved problem in mathematics , 

36 


of months ago when a sudden flash of insight struck me 
and a kind of luminous glow irradiated the interior of my 
skull. / was able to prove the truth of Fermat's Last 
Theorem in a very simple way . 

When I tell you that the greatest mathematicians of the 
last three centuries have tackled Fermat’s Last Theorem 
with ever increasingly sophisticated mathematical tools 
and that all have failed, you will realize what a stroke of 
unparalleled genius it was for me to succeed with nothing 
more than ordinary arithmetical reasoning. 

My delirium of ecstasy did not completely blind me to 
the fact that my proof depended upon one assumption 
which I could check very easily with pencil and paper. I 
went upstairs to my study to carry that check through — 
stepping very carefully so as not to jar all that brilliance 
inside my cranium. 

You guessed it, I’m sure. My assumption proved to be 
quite false inside of a few minutes. Fermat’s Last Theorem 
was not proven after all; and my radiance paled into the 
light of ordinary day as I sat at my desk, disappointed 
and miserable. 

Now that I have recovered completely, however, I look 
back on that episode with some satisfaction. After all, for 
five minutes, I was convinced that I was soon to be recog- 
nized as the most famous living mathematician in the 
world, and words cannot express how wonderful that felt 
while it lasted! 

On the whole, though, I suppose that true old findings, 
however minor, are better than new false ones, however 
major. So I will trot out for your delectation, a little dis- 
covery of mine which I made just the other day but which, 
I am certain, is over three centuries old in reality. 

However, I’ve never seen it anywhere, so until some 
Gentle Reader writes to tell me who first pointed it out 
and when, I will adopt the discovery as the Asimov Series. 

First, let me lay the groundwork. 

We can begin with the following expression; (l + l/n) n 
where n can be set equal to any whole number. Suppose 
we try out a few numbers. 

If n— 1, the expression becomes ( 1 + Vi ) 1 = 2, If n=2, 
the expression becomes ( 1 + Vs) 2 or (%>) 2 or % or 2.25. If 
n— 3, the expression becomes (1++) 3 or (¥sV or u k~ or 
about 2.3704. 



38 


NUMBERS AND COUNTING 


Exclamation Point! 


39 


We can prepare Table 1 of the value of the expression 
for a selection of various values of n: 

Table 1 The Approach to e 
n (l + l/n) n 

1 2 


2 

2.25 

3 

2.3704 

4 

2.4414 

5 

2.4888 

10 

2.5936 

20 

2.6534 

50 

2.6915 

100 

2.7051 

200 

2.7164 


my library — and those aren’t accurate enough to handle 
values of n over 200 in this case. In fact, I don't trust my 
value for n— 200. 

Fortunately, there are other ways of determining e . 
Consider the following series: 2 + :£+ % +V>4 +H 20 + 
¥120 * * * 

There are six members in this scries of numbers as far 
as I've given it above, and the successive sums are: 

2 

2.5 

2.6666 . . . 

2.7083333 . . . 
2.7166666 . . . 
2.71805555 . . , 




hi — — 

2 + ^ + /6 + + 7120 “ 


As you see, the higher the value of n t the higher the 
value of the expression (I + l/n) n . Nevertheless, the value 
of the expression increases more and more slowly as n 
increases. When n doubles from 1 to 2, the expression in- 
creases in value by 0.25. When n doubles from 100 to 200, 
the expression increases in value only by 0.0113. 

The successive values of the expression form a “con- 
verging series” which reaches a definite limiting value. 
That is, the higher the value of n } the closer the value of 
the expression comes to a particular limiting value without 
ever quite reaching it (let alone getting past it). 

The limiting value of the expression (l + l/n) n as n 
grows larger without limit turns out to be an unending 
decimal, which is conventionally represented by the sym- 
bol e. 

It so happens that the quantity e is extremely important 
to mathematicians and they have made use of computers 
to calculate its value to thousands of decimal places. Shall 
we make do with 50? AH right. The value of e is: 2.7182- 
8182845904523536028747135266249775724709369995... 

You may wonder how mathematicians compute the 
limit of the expression to so many decimal places. Even 
when I carried n up to 200 and solved for (l+^oo) 200 , I 
only got e correct to two decimal places. Nor can I reach 
higher values of rc. I solved the equation for n=200 by 
the use of five-place logarithm tables — the best available in 


In other words, by a simple addition of six numbers, a 
process for which I don't need a table of logarithms at 
all, I worked out e correct to three decimal places. 

If I add a seventh number in the scries, then an eighth, 
and so on, I could obtain e correct to a surprising number 
of additional decimal places. Indeed, the computer which 
obtained the value of e to thousands of places made use 
of the series above, summing thousands of fractions in the 
series. 

But how does one tell what the next fraction in the 
series will be? In a useful mathematical series, there 
should be some way of predicting every member of the 
series from the first few. If I began a series as follows: 
Vi+Vi+M+V, . . . you would, without trouble continue 
onward . Vj+Vr+H . * . Similarly, if a series began 
you would be confident in continuing . . . 
^2 4- Hi 4 + Vi 28 . * ■ 

In fact, an interesting parlor game for number-minded 
individuals would be to start a series and then ask for the 
next number. As simple examples consider: 

2,3,5, 7, 1 1 . . . 

2, 8, 18, 32, 50 . . . 

Since the first series is the list of primes, the next num- 
ber is obviously 13, Since the second series consists of 
numbers that are twice the list of successive squares, the 
next number is 72. 



40 


NUMBERS AND COUNTING 


Exclamation Point / 


41 


But what are we going to do with a series such as: 

2 +H+H+H 4 +H 2 Q+H 20 . . . What is the next number? 

If you know, the answer is obvious, but if you hadn't 
known, would you have been able to see it? And if you 
don't know, can you see it? 

Just briefly, I am going to introduce a drastic change of 
subject. 

Did any of you ever read Dorothy Sayers* Nine Tailors ? 
I did, many years ago. It is a murder mystery, but I re- 
member nothing of the murder, of the characters, of the 
action, of anything at all but for one item. That one item 
involves “ringing the changes,” 

Apparently (I slowly gathered as I read the book) in 
ringing the changes, you begin with a series of bells tuned 
to ring different notes, with one man at the rope of each 
bell. The bells are pulled in order: do, re, mi, fa, and so 
on. Then, they are pulled again, in a different order. Then, 
they are pulled again in a still different order. Then, they 
are pulled again — 

You keep it up until all the possible orders (or 
“changes”) in which the bells may be rung are rung. One 
must follow certain rules in doing so, such that no one 
bell, for instance, can be shifted more than one unit out 
of its place in the previous change. There are different 
patterns of shifting the order in the various kinds of 
change-ringing and these patterns are interesting in them- 
selves, However, all I am dealing with here arc the total 
number of possible changes connected with a fixed num- 
ber of bells. 

Let’s symbolize a bell by an exclamation point (!) to 
represent its clapper, so that we can speak of one bell as 
1!, two bells as 2! and so on. 

No bells at all can be rung in one way only — by not 
ringing — so 0!— I. One bell (assuming bells must be rung 
if they exist at all) can only be rung in one way— be ng — ■ 
so I! = l. Two bells, a and /a can dearly be rung in two 
ways, ah and da r -o 2!-; 2. 

Three bells, a, Ik and can be rung in six ways: abc f 
deb. hac, be a, cab , and cha, and no more, so 2 ! — 6 . Four 
bells, c.% b t and a, can be rung in lust twenty-four dif- 
ferent ways. I won't list them all, but you can start with 
abedt abdc, acbd , and aedb and see how many more 
changes you can list. If vgu can list twenty- five different 


and distinct orders of writing four letters, you have shaken 
the very foundations of mathematics, but I don’t expect 
you will be able to do it. Anyway, 4!=24. 

Similarly (take my word for it for just a moment), five 
bells can be rung in 120 different changes and six bells in 
720, so that 5!=120 and 6!— 720. 

By now I think you’ve caught on. Suppose we look 
again at the series that gives us our value of e: 2+%+ 
H-J-bk+Hao +H 20 . . . and write it this way: 
e=%,+M l +% 1 +% l +% ! +% ! +%: ... 

Now we know how to generate the fractions next in 
line. They are . . . H-H.-J-H.+Ht and so on forever. 

To find the values of fractions such as W T , and Hi, 
you must know the value of 7!, 81, and 9! and to know 
that you must figure out the number of changes in a set 
of seven bells, eight bells, and nine bells. 

Of course, if you’re going to try to list all possible 
changes and count them, you’ll be at it all day; and you’ll 
get hot and confused besides. 

Let’s search for a more indirect method, therefore. 

We’ll begin with four bells, because fewer bells offer no 
problem. Which bell shall we ring first? Any of the four, 
of course, so we have four choices for first place. For 
each one of these four choices, we can choose any of 
three bells (any one, that is, except the one already chosen 
for first place) so that for the first two places in line we 
have 4x3 possibilities. For each of these we can choose 
either of the two remaining bells for third place, so that 
for the first three places, we have 4x3x2 possibilities. 
For each of these possibilities there remains only one bell 
for fourth place, so for all four places there are 4x3x2 
X 1 arrangements. 

Wc can say then, that 41=4x3x2x1=24. 

If we work out the changes for any number of bells, 
we will reach similar conclusions. For seven bells, for in- 
stance, the total number of changes is 7x6x5x4x3x2 
X 1=5,040. We can say, then, that 7! = 5.040. 

(The common number of bells used in ringing the 
changes is seven; a set termed a “peal,” If all seven bells 
are rung through once in six seconds, then a complete set 
of changes — 5,040 of them — requires eight hours, twenty- 
four minutes . . . And ideally, it should be done without 
a mistake. Ringing the changes is a serious thing.) 



42 


NUMBERS AND COUNTING 


Exclamation Point! 


43 


CHURCH BELLS 

Bells, which l use to illustrate factorial numbers in this 
essay , are common to a wide variety of cultures . In our 
own , they are most associated with churches f and in the 
days before modern timepieces } they were the universal 
method of apprising the population of the time f calling 
people to prayers , for instance. (I was in Oxford , Eng- 
land, one Sunday morning in 1974 when the bells started 
pealing — and kept on pealing. The din was indescribable 
and as Robert Heinlein once said, “If a nightclub made 
half that much noise they would shut it down as a public 
nuisance.”) 

Bells were also used to sound the alarm in case of fire , 
of an enemy approach , and so on. They were also rung 
during thunderstorms to keep off the lightning . Since 
church towers are usually the tallest structures in the 
towns of medieval and early modern towers , they were 
often struck by lightning and the bell ringing did nothing 
to prevent it. In fact , many bell ringers were killed by 
lightning . 

As for change ringing, the “nine tailors " / mention in 
the article is by no means maximum , As many as twelve 
bells are used in change ringing , and changes rung on that 
many bells is called a “maximus” 

Well it might be, since nothing but partial changes can 
be rung on twelve bells. A complete change in which 
every possible permutation of twelve bells is carried 
through in order would involve 479,001,600 different 
ringings of each bell . Where a ** minimus ,** involving four 
bells , can be put through a complete change in thirty 
seconds T a “ maximus ** would require about forty years ! 

Change ringing is associated particularly with the 
Church of England and was originally a gentleman's 
recreation. Thus , Lord Peter Wimsey pulls a mean bell in 
The Nine Tailors. 


Culver Pictures t Inc . 



44 


NUMBERS AND COUNTING 


Exclamation Point! 


45 


Actually, the symbol does not really mean “belt” 
(That was just an ingenious device of mine to introduce 
the matter.) In this case it stands for the word “factorial.” 
Thus, 4! is “factorial four” and 7! is “factorial seven.” 

Such numbers represent not only changes that can be 
rung in a set of bells, but the number of orders in which 
the cards can be found in a shuffled deck, the number of 
orders in which men can be seated at a table, and so on. 

I have never seen any explanation for the term “fac- 
torial 1 ’ but I can make what seems to me a reasonable stab 
at explaining it. Since the number 5,040—7x6x5x4x3 
X2xl, it can be evenly divided by each number from 1 
to 7 inclusive. In other words, each number from 1 to 7 
is a factor of 5,040; why not, therefore, call 5,040, “fac- 
torial seven/ 1 

And we can make it general. All the integers from 1 to 
n are factors of nU Why not call n! “factorial n*' therefore. 

We can see, now, why the series used to determine e 
is such a good one to use. 

The values of the factorial numbers increase at a tre- 
mendous rate, as is clear from the list in Table 2 of values 
up to merely 15! 


Table 2 

The Factorials 

0! 

1 

1! 

1 

2! 

2 

3! 

6 

4! 

24 

5! 

120 

6! 

720 

7! 

5,040 

8! 

40,320 

9! 

362,880 

10! 

3,628,800 

11! 

39,916,800 

12! 

479,001,600 

13! 

6,227,020,800 

14! 

87,178,291,200 

15! 

1,307,674,368.000 


As the values of the factorials zoom upward, the value 
of fractions with successive factorials in the denominator 


must zoom downward. By the time you reach Vg : , the value 
is only # 20 , and by the time you reach Vis-, the value is 
considerably less than a trillionth. 

Each such factorial d enominato red fraction is larger 
than the remainder of the series all put together. Thus 
Yi^i is larger than Vi^+ViTr-f Via- . . . and so on and so on 
forever, all put together. And this preponderance of a 
particular fraction over all later fractions combined in- 
creases as one goes along the series. 

Therefore suppose we add up all the terms of the series 
through Vu-. The value is short of the truth by V / i 5 :+M<j!+ 
ViTs+Vis. etc, etc. We might, however, say the value is 
short of the truth by Vir> : because the remainder of the 
series is insignificant in sum compared to Vis-. The value 
of V4 e. is less than a trillionth. It is, in other words, less 
than 0.000000000001, and the value of e you obtain by 
summing a little over a dozen fractions is correct to eleven 
decimal places. 

Suppose we summed all the series up to V&W: (by com- 
puter, of course). If we do that, we are Viooo ; short of the 
true answer. To find out how much that is, we must have 
some idea of the value of 1000!. We might determine 
that by calculating 1000x999x998 ... and so on, but 
don't try. It will take forever. 

Fortunately, there exist formulas for calculating out 
large factorials (at least approximately) and there are 
tables which give the logarithms of these large factorials. 

Thus, log 1000! = 2567.6046442. This means that 1000! 
=4.024 x 10 2567 , or (approximately) a 4 followed by 
2,567 zeroes. If the series for e is calculated out to Vto:, 
the value will be short of the truth by only 1/ (4x iO 2567 ) 
and you will have e correct to 2,566 decimal places. (The 
best value of e I know of was calculated out to no less 
than 60,000 decimal places.) 

Let me digress once again to recall a time I had per- 
sonal use for moderately large factorials. When I was in 
the Army, I went through a period where three fellow 
sufferers and myself played bridge day and night until 
one of the others broke up the thing by throwing down 
his hand and saying, “We’ve played so many games, the 
same hands are beginning to show up.” 

I was terribly thankful, for that gave me something to 
think about. 



46 


NUMBERS AND COUNTING 


Exclamation Point ! 


47 


Each order of the cards in a bridge deck means a pos- 
sible different set of bridge hands. Since there are fifty-two 
cards, the total number of arrangements is 52!. However, 
within any individual hand, the arrangement doesn't mat- 
ter. A particular set of thirteen cards received by a par- 
ticular player is the same hand whatever its arrangement. 
The total number of arrangements of the thirteen cards 
of a hand is 13! and this is true for each of four hands. 
Therefore the total number of bridge-hand combinations 
is equal to the total number of arrangements divided by 
the number of those arrangements that don't matter, or: 

52! 

( 1 3! ) 4 

I had no tables handy, so I worked it out the long way 
but that didn’t bother me. It took up my time and, for my 
particular tastes, was much better than a game of bridge. 
I have lost the original figures long since, but now I can 
repeat the work with the help of tables. 

The value of 52! is, approximately, 8.066xl0 6T . The 
value of 13! (as you can see in the table of factorials I 
gave above) is approximately 6.227x10° and the fourth 
power of that value is about 1.5 XlO 33 . If we divide 
8.066X10 67 by 1.5 xlO 33 , we find that the total number 
of different bridge games possible is roughly 5.4xl0 £8 or 
54,000,000,000,000,000,000,000,000,000 or 54 octillion. 

I announced this to my friends. I said, “The chances 
are not likely that we are repeating games. We could play 
a trillion games a second for a billion years, without re- 
peating a single game.” 

My reward was complete incredulity. The friend who 
had originally complained said, gently, “But, pal, there are 
only fifty-two cards, you know,” and he led me to a quiet 
corner of the barracks and told me to sit and rest awhile. 

Actually, the series used to determine the value of e is 
only a special example of a general case. It is possible to 
show that : 

e*=x 0 /0!+x71!+x-/2!+x 3 /3!+xV4!+xV5! . . . 

Since jc° = 1, for any value of x , and 0! and 1! both equal 
1, the series is usually said to start: e*— l+x+x 2 /2!+x 3 /3! 


, . , but I prefer my version given above. It is more sym- 
metrical and beautiful. 

Now e itself can be expressed as e 1 . In this case, the x 
of the general series becomes 1. Since 1 to any power 
equals 1, then x 2 , jc 3 , jc 4 and all the rest become 1 and the 
series becomes: 

e 1 =14 +14; +14: + 14+ 14; +14 . . . which is just the series 
we’ve been working with earlier. 

But now let’s take up the reciprocal of e\ or, in other 
words, 1 / e . Its value to fifteen decimal places is 0.367879- 
441171442 ... 

It so happens that \/e can be written as which 
means that in the general formula for c x , we can substitute 

— 1 for x. 

When — 1 is raised to a power, the answer is +1 if the 
power is an even one, and — 1 if it is an odd one. In other 
words: (-1)«=1, (-lj^-l, (-1) 2 = + 1, (-l) s = 

— 1, ( — 1) 4 = + 1, and so on forever. 

If, in the general series, then, x is set equal to —1, we 
have: 

e - 1 =(-l)V0! + (-l) 1 /l!+(-l)V2! + (-l)V3! + 
(_1)V4! . . .or e^ = l/0! + (-l)/l!+14!+(-l)/3!+14! 
+ ( — 1)/5! . . . or e -1 — 14—14+14 — 14+ 14 : — 14 + 

%-Vtt ... 

In other words, the series for 1/e is just like the series 
for e except that all the even terms are converted from 
additions to subtractions. 

Furthermore, since 1/0! and 14 both equal 1, the first 
two terms in the series for 1/e— 1/0!— 14 — are equal to 
1 — 1 “0. They may therefore be omitted and we may 
conclude that 

e _1 =14 — + +4t — 14+14 — 14+14^ 14+14 o>, and so on 

forever. 

And now, at last, we come to my own personal discov- 
ery! As I looked at the series just given above for e” 1 , I 
couldn’t help think that the alternation between plus and 
minus is a flaw in its beauty. Could there not be any way 
in which it could be expressed with pluses only or with 
minuses only? 

Since an expression such as —14! +14! can be converted 
into— (14!— 14!), it seemed to me I could write the follow- 
ing series: 

e" 1 =%,— (% : — % T ) — ^ (14-14) ... and so 

on. 



48 


NUMBERS AND COUNTING 


PLAYING CARDS 

It is because of the rapid increase in the factorial num- 
bers that it is possible to play an infinite number of games 
(infinite with respect to the limited human life span) with 
a mere fifty-two cards. 

Indeed , the only other common deck is the pinochle 
deck in which there are only the ace, king , queen f jack, 
ten , and nine , and where each suit is duplicated . With 
eight kinds of each of six kinds of cards t you have only 
forty-eight cards . This involves a lower factorial, and the 
duplication of suits also cuts into the number of different 
hands possible. This means there are only 1/312,000,000 
times as many different hands with a pinochle deck as 
with an ordinary deck, but the smaller number is still 
enough to supply no fear of duplications in the course 
of dedicated playing of the game of pinochle. 

Somehow one gets the feeling that card games , which 
are so ubiquitous in the present-day world, must be an 
ancient, even a prehistoric pastime, but not so. They are 
a medieval invention, probably originating in the Far East 
and reaching Europe in the 1200s . They may have been 
brought west by Marco Polo or by gypsies or by Arab 
conquerors; no one really knows. 

Odder still, two of the properties of playing cards, 
which we take quite for granted now, are even more 
recent modifications. One is the presence of the small index 
in the upper left and lower right corners so that we can 
identify a card when only a small part is exposed, as in 
the illustration. The other is the up-and-down rotational 
symmetry so that the card is right-side-up either way . If 
you were to try to play cards without these modifications, 
you would be appalled by the inconvenience. 

Cards , incidentally, may have been used for fortune- 
telling (the tarot deck) before they were used for games 
of chance . 



Fundamental Photographs from the Granger Collection 




50 


NUMBERS AND COUNTING 


Exclamation Point! 


51 


Now we have only minus signs, but we also have paren- 
theses, which again offer an aesthetic flaw. 

So I considered the contents of the parentheses. The first 
one contains V:; : — Vi; which equals 1/(3 x2x I ) — 1 / (4x3 X 
2X1). This is equal to (4— 1 ) / ( 1 x3 x2x 1), or to %. 
In the same way, *4— %■ = #,--%=%, and so on. 

I was astonished and inexpressibly delighted for now I 
had the Asimov Series which goes: 

. . . and so on forever. 

I am certain that this series is at once obvious to any 
real mathematician and Fm sure it has been described in 
texts for three hundred years — but I’ve never seen it and 
until someone stops me, I’m calling it the Asimov Series. 

Not only does the Asimov Series contain only minus 
signs (except for the unexpressed positive sign before the 
first term), but it contains all the digits in order. You 
simply can’t ask for anything more beautiful than that. 
Let’s conclude now, by working out just a few terms of 
the series: 

~0.5 

=0.375 

94i— 56, =0.3680555 . . . 

=0.3678819 . . . 

As you see, by adding up only four terms of the series, 
I get an answer which is only 0.0000025 greater than the 
truth, an error of 1 part in a bit less than 150,000 or, 
roughly Vi:m of 1 per cent. 

So if you think the ‘'Exclamation Point” of the title 
refers only to the factorial symbol, you are wrong. It ap- 
plies even more so to my pleasure and astonishment with 
the Asimov Series. 

p.s. To get round the unexpressed positive sign in the 
Asimov Series some readers (after the first appearance in 
print of this chapter) suggested the series be written: 

— ( — l)/ot— 34 . ... All the terms would then indeed 
be negative, even the first, but we would have to step out- 
side the realm of the natural numbers to include 0 and 

— 1, which detracts a bit from the austere beauty of the 
series. 

Another suggested alternative is: 

. . . which also gives l/e. It includes only positive signs 


which are prettier (in my opinion) than negative signs 
but, on the other hand, it includes 0. 

Still another reader suggested a similar series for e itself; 
one that goes as follows: 9i +% . . The 

inversion of the order of the natural numbers detracts 
from its orderliness but it gives it a certain touch of charm- 
ing grace, doesn’t it? 

Oh, if only mathematics loved me as I love her! 



T-Formation 


53 


A 

T-FORMATION 


I have been accused of having a mad passion for 
targe numbers and this is perfectly true. I wouldn't dream 
of denying it. However, may I point out that I am not the 
only one? 

For instance, in a book entitled Mathematics and the 
Imagination (published in 1940) the authors, Edw T ard Kas- 
ner and James Newman, introduced a number called the 
“googoL” which is good and large and which was promptly 
taken up by writers of books and articles on popular 
mathematics- 

Per son ally, I think it is an awful name, but the young 
child of one of the authors invented it, and what could a 
proud father do? Thus, we are afflicted forever with that 
baby-talk number. 

The googol was defined as the number 1 followed by a 
hundred zeros, and so here (unless I have miscounted or 
the Noble Printer has goofed) is the googol, written out 
in full: 

10 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 

000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 000 , 

000,000,000,000,000,000,000, 

Now this is a pretty clumsy way of writing a googol, 
but it fits in w j ith our system of numeration, which is 
based on the number 10. To write large numbers we sim- 

52 


ply multiply 10’s, so that a hundred is ten times ten and is 
written 100; a thousand is ten times ten times ten and is 
written 1000 and so on. The number of zeros in the num- 
ber is equal to the number of tens being multiplied, so 
that the googol, with a hundred zeros following the 1, is 
equal to a hundred tens multiplied together. This can also 
be written as 10 100 . And since 100 is ten times ten or 10 2 , 

the googol can even be written as 10 10 ' 

Certainly, this form of exponential notation (the little 
figure in the upper right of such a number is an ex- 
ponent”) is very convenient, and any book on popular 
math will define a googol as 10 100 . However, to anyone 
who loves large numbers, the googol is only the beginning 
and even this shortened version of writing large numbers 
isn't simple enough.* 

So I have made up my own system for writing large 
numbers and 1 am going to use this chapter as a chance 
to explain it. (Freeze, everyone! No one’s leaving till I’m 

through.) . ^ 

The trouble, it seems to me, is that we are using the 
number 10 to build upon. That was good enough for cave 
men, I suppose, but we moderns are terribly sophisticated 
and we know lots better numbers than that. 

For instance, the annual budget of the United States of 
America is in the neighborhood, now, of $100,000,000,000 
(a hundred billion dollars). That means 1,000,000,000,- 
000 (one trillion) dimes.** 

Why don’t we, then, use the number, one trillion, as a 
base? To be sure, we can’t visualize a trillion, but why 
should that stop us? We can't even visualize fifty-three. 
At least if someone were to show us a group of objects 
and tell us there are fifty-three of them altogether, we 
couldn’t tell whether he were right or wrong without count- 
ing them. That makes a trillion no less unreal than fifty- 
three, for we have to count both numbers and both are 
equally countable. To be sure, it would take us much 
longer to count one trillion than to count fifty-three, but 


* The proper name for the googol, using American nomenclature, is " ten 
duotrigintilUon > ,> but I dare say, gloomily, that that will never replace 

[** This article first appeared in August 1965. Since then the budget has 
more than tripled and is over three trillion dimes. Aren’t u’e lucky? In - 
cidentally , footnotes added for this collection are in brackets to distinguish 
them from those present at the time of first appearance A 



54 


NUMBERS AND COUNTING 


7-Formation 


55 


the principle is the same and I, as anyone will tell you, am 
a man of principle. 

The important thing is to associate a number with some- 
thing physical that can be grasped and this we have done. 
The number 1,000,000,000,000 is roughly equal to the 
number of dimes taken from your pocket and mine 
(mostly mine, I sometimes sullenly think) each year by 
kindly, jovial Uncle Sam to build missiles and otherwise 
run the government and the country. 

Then, once we have it firmly fixed in our mind as to 
what a trillion is, it takes very little effort of imagination 
to see what a trillion trillion is; a trillion trillion trillion, 
and so on. In order to keep from drowning in a stutter 
of trillions, let’s use an abbreviated system that, as far as 
I know, is original with me.* 

Let’s call a trillion T-l; a trillion trillion T-2; a trillion 
trillion trillion T-3, and form large numbers in this fash- 
ion. (And there’s the “T-formation” of the title! Surely 
you didn’t expect football?) 

Shall we see how these numbers can be put to use? I 
have already said that T-l is the number of dimes it takes 
to run the United States for one year, In that case, T-2 
would represent the number of dimes it would take to run 
the United States for a trillion years. Since this length of 
time is undoubtedly longer than the United States will 
endure (if I may be permitted this unpatriotic sentiment) 
and, in all likelihood, longer than the planet earth will 
endure, we see that we have run out of financial applica- 
tions of the Asimovian T-numbers long before we have 
even reached T-2, 

Let’s try something else. The mass of any object is 
proportional to its content of protons and neutrons which, 
together, may be referred to as nucleons. Now T-l nu- 
cleons make up a quantity of mass far too small to see in 
even the best optical microscope and even T-2 nucleons 
make up only 1 % grams of mass, or about Vic of an ounce. 

Now we’ve got room, it would seem, to move way up 
the T-scale. How massive, for instance, are T-3 nucleons? 
Since T-3 is a trillion times as large as T-2, T-3 nucleons 


* Actually, Archimedes set up a system of numbers based on the myriad, 
and spoke of a myriad myriad, a myriad myriad myriad and so on. But a 
myriad is only 10,000 and I’m using 1,000, 000,000,000, so I don’t con- 
sider Archimedes to be affecting my originality. Besides, he only beat me 
out by less than twenty-two centuries. 


have a mass of 1.67 trillion grams, or a little under two 
million tons. Maybe there’s not as much room as we 
thought. 

In fact, the T-numbers build up with breath-taking 
speed. T-4 nucleons equal the mass of all the earth’s 
ocean, and T~5 nucleons equal the mass of a thousand 
solar systems. If we insist on continuing upward. T-6 nu- 
cleons equal the mass of ten thousand galaxies the size of 
ours, and T-7 nucleons are far, far more massive than the 
entire known universe. 

Nucleons are not the only subatomic particles there are, 
of course, but even if we throw in electrons, mesons, neu- 
trinos, and all the other paraphernalia of subatomic struc- 
ture, we cannot reach T-7. In short, there are far less than 
T-7 subatomic particles of all sorts in the visible universe. 

Clearly, the system of T-numbers is a powerful method 
of expressing large numbers. How does it work for the 
googol? Well, consider the method of converting ordinary 
exponential numbers into T-numbers and vice versa. T-l 
is equal to a trillion, or 10 13 ; T-2 is equal to a trillion tril- 
lion, or 10- 1 , and so on. Well, then, you need only divide 
an exponent by 12 to have the numerical portion of a 
T-number; and you need only multiply the numerical por- 
tion of a T-number by 12 to get a ten-based exponent. 

If a googol is 10 10( \ then divide 100 by 12, and you see 
at once that it can be expressed as T-8V2. Notice that 
T-8 1 /^ is larger than T-7 and T-7 is in turn far larger than 
the number of subatomic particles in the known universe. 
It would take a billion trillion universes like our own to 
contain a googol of subatomic particles. 

What then is the good of googol, if it is too large to be 
useful in counting even the smallest material objects spread 
through the largest known volume? 

I could answer: For its own sheer, abstract beauty — 

But then you would all throw rocks at me. Instead, then, 
let me say that there are more things to be counted in this 
universe than material objects. 

For instance, consider an ordinary deck of playing 
cards. In order to play, you shuffle the deck, the cards 
fall into a certain order, and you deal a game. Into how 
many different orders can the deck be shuffled? (Since it 
is impossible to have more essentially different game- 
situations than there are orders-of-cards in a shuffled deck, 



56 


NUMBERS AND COUNTING 


T -For mat ion 


57 


this is a question that should interest your friendly neigh- 
borhood poker-player.) 

The answer is easily found (see Chapter 3) and comes 
out to about 80,000,000,000,000,000,000,000,000,000,000, 
000,000,000,000,000,000,000,000,000,000,000,000, or 8X 
10 6T . In T-numbers, this is something like T -5%. With an 
ordinary deck of cards, then, we can count arrangements 
and reach a value equal to that of the number of sub- 
atomic particles in a galaxy, more or less. 

If, instead of 52 cards, we played with 70 cards (and 
this is not unreasonable; canasta, I understand, uses 108 
cards), then the number of different orders after shuffling, 
just tops the googol mark. 

So when it comes to analyzing card games (let alone 
chess, economics, and nuclear war), numbers like the 
googol and beyond are met with. 

Mathematicians, in fact, are interested in many varieties 
of numbers (with and without practical applications) in 
which vastnesses far, far beyond the googol are quickly 
reached. 

Consider Leonardo Fibonacci, for instance, the most ac- 
complished mathematician of the Middle Ages. (He was 
born in Pisa, so he is often called Leonardo of Pisa.) 
About 1200, when Fibonacci was in his prime, Pisa was 
a great commercial city, engaged in commerce with the 
Moors in North Africa. Leonardo had a chance to visit 
that region and profit from a Moorish education. 

The Moslem world had by that time learned of a new 
system of numeration from the Hindus. Fibonacci picked 
it up and in a book, Liber Abaci , published in 1202, in- 
troduced these ‘‘Arabic numerals 11 and passed them on to 
a Europe still suffering under the barbarism of the Roman 
numerals (see Chapter 1). Since Arabic numerals are only 
about a trillion times as useful as Roman numerals, it took 
a mere couple of centuries to convince European mer- 
chants to make the change. 

In this same book Fibonacci introduces the following 
problem: “How many rabbits can be produced from a sin- 
gle pair in a year if every month each pair begets a new 
pair, which from the second month on become productive, 
and no deaths occur?” (It is also assumed that each pair 
consists of a male and female and that rabbits have no 
objection to incest) 


In the first month we begin with a pair of immature 
rabbits, and in the second month we still have one pair, 
but now they are mature. By the third month they have 
produced a new pair, so there are two pairs, one mature, 
one immature. By the fourth month the immature pair 
has become mature and the first pair has produced another 
immature pair, so there are three pairs, two mature and 
one immature. 

You can go on if you wish, reasoning out how many 
pairs of rabbits there will be each month, but I will give 
you the series of numbers right now and save you the 
trouble. It is: 

1, 1,2, 3, 5, 8, 13, 21,34, 55, 89, 144 

At the end of the year, you see, there would be 144 
pairs of rabbits and that is the answer to Fibonacci s 
problem. 

The series of numbers evolved out of the problem is the 
“Fibonacci series' 1 and the individual numbers of the series 
are the “Fibonacci numbers.” If you look at the series, 
you will see that each number (from the third member 
on) is the sum of the two preceding numbers. 

This means we needn't stop the series at the twelfth 
Fibonacci number (F n ). We can construct F ia easily 
enough by adding F u and F 12 , Since 89 and 144 are 233, 
that is F 13 . Adding 144 and 233 gives us 377 or F 14 . We 
can continue with F l5 equal to 610, F JC equal to 987, and 
so on for as far as we care to go. Simple arithmetic, noth- 
ing more than addition, will give us all the Fibonacci 
numbers we want. 

To be sure, the process gets tedious after a while as the 
Fibonacci numbers stretch into more and more digits and 
the chances of arithmetical error increase. One arithmetical 
error anywhere in the series, if uncorrected, throw's off all 
the later members of the series. 

But why should anyone want to carry the Fibonacci 
sequence on and on and on into large numbers? Well, the 
series has its applications. It is connected with cumulative 
growth, as the rabbit problem shows, and, as a matter of 
fact, the distribution of leaves spirally about a lengthening 
stem, the scales distributed about a pine cone, the seeds 
distributed in the sunflower center, all have an arrange- 
ment related to the Fibonacci series. The series is also re- 



58 


NUMBERS AND COUNTING 


T-Formation 


59 


la ted to the "golden section, 14 which is important to art 
and aesthetics as well as to mathematics. 

But beyond all that, there are always people who are 
fascinated by large numbers. (I can’t explain the fascina- 
tion but believe me it exists.) And if fascination falls short 
of working away night after night with pen and ink, it is 
possible, these days, to program a computer to do the 
work, and get large numbers that it would be impractical 
to try to work out in the old -fashioned way. 

The October 1962 issue of Recreational Mathematics 
Magazine * lists the first 571 Fibonacci numbers as worked 
out on an IBM 7090 computer. The fifty-fifth Fibonacci 
number passes the trillion mark, so that we can say that 
F 55 is greater than T-t. 

From that point on, every interval of fifty-five or so 
Fibonacci numbers (the interval slowly lengthens) passes 
another T-number. Indeed, F 4S1 is larger than a googol. It 
is equal to almost one and a half googols, in fact. 

Those multiplying rabbits, in other words, will quickly 
surpass any conceivable device to encourage their multi- 
plication. They will outrun any food supply that can be 
dreamed up, any room that can be imagined. There might 
be only 144 at the end of a year, but there would be nearly 
50,000 at the end of two years, 15,000,000 at the end of 
three years, and so on. In thirty years there would be more 
rabbits than there are subatomic particles in the known 
universe, and in forty years there would be more than a 
googol of rabbits. 

To be sure, human beings do not multiply as quickly 
as Fibonacci's rabbits, and old human beings do die. 
Nevertheless, the principle remains. What those rabbits 
can do in a few years, we can do in a few centuries or 
millenniums. Soon enough. Think of that when you tend 
to minimize the population explosion. 

For the fun of it, I would like to write F 571 , which is the 
largest number given in the chapter. (There will be larger 
numbers later, but I will not write them out!) Anyway, 
F*„ is: 96041200618922553823942883360924865026104 
917411877067816822264789029014378308478864192589 


* This is a fascinating little periodical which I heartily recommend to 
any nut congruent to myself . 


084185254331637646183008074629. This vast number is 
not quite equal to T-10.* 

For another example of large numbers, consider the 
primes. These are numbers like 7, or 641, or 5237, which 
can be divided evenly only by themselves and 1 . They have 
no other factors. You might suppose that as one goes 
higher and higher in the scale of numbers, the primes 
gradually peter out because there would be more and more 
smaller numbers to serve as possible factors. 

This, however, does not happen, and even the ancient 
Greeks knew that. Euclid was able to prove quite simply 
that if all the primes are listed up to a “largest prime, it 
is always possible to construct a still larger number which 
is either prime itself or has a prime factor that is larger 
than the “largest prime.” It follows then there is no such 
thing as a “largest prime” and the number of primes is 
infinite. 

Yet even if we can’t work out a largest prime, there is 
an allied problem. What is the largest prime we know? It 
would be pleasant to point to a large number and say: 
“This is a prime. There are an infinite number of larger 
primes, but we don’t know which numbers they are. This 
is the largest number we know to be a prime.” 

Once that is done, you see, then some venturesome 
amateur mathematician may find a still larger prime. 

Finding a really large prime is by no means easy. 
Earlier, for instance, I said that 5237 is prime. Suppose 
you doubted that, how would you check me? The only 
practical way is to try all the prime numbers smaller than 
the square root of 5237 and see which, if any, are factors. 
This is tedious but possible for 5237. It is simply imprac- 
tical for really large numbers — except for computers. 

Mathematicians have sought formulas, therefore, that 
would construct primes. It might not give them every 
prime in the book, so that it could not be used to test a 
given number for prime-hood. However, it could construct 
primes of any desired size, and after that the task of 


* Since this was written, the editor of Recreational Mathematics wrote io 
say that he had new Fibonacci numbers, up to Fiooo- This Fiooo with 209 
digits, is something over T-17. lEteven years have passed since this foot- 
note first appeared, but I have heard nothing more. I'm sure that new 
Fibonacci numbers have been worked out but, alas , it is difficult to keep 
up with everything. Things get past me .] 



60 


NUMBERS AND COUNTING 


LEONARDO FIBONACCI 

Leonardo Fibonacci was born in Pisa about 1170 and he 
died about 1230 . As 7 said in this essay , his greatest 
achievement was in popularizing the Arabic numerals in 
his book liber Abaci, In this , he had been anticipated 
by the English scholar Adelard of Bath ( tutor of Henry ll 
before that prince had succeeded to the throne) a century 
earlier . It was Fibonacci's book, however , that made the 
necessary impression* 

But why did he call it Liber Abaci, or Book of the 
Abacus? Because, oddly enough > the use of Arabic nu- 
merals was implicit in the “abacus” a computing device 
that dates back to Babylonia and the earliest days of 
history . 

The abacus, in its simplest form, is most easily visualized 
as a series of wires on each of which ten counters are 
strung . There is room on the wire to move one or more 
of the counters to the right or left t 

If you want to add five and four , for instance , you 
move five counters leftward , then four more , and count 
all you have moved — nine. If you want to add five and 
eight, you move five counters , but only have five more , 
not eight more . to move . You move the five , convert the 
ten counters into one counter in the wire above , then move 
the remaining three. The counters in the wire above are 
“ tens T " so you have one ten and three ones for a total 
of thirteen. 

The wires represent, successively , units, tens , hundreds . 
thousands, and so on, and Arabic numerals, in essence, 
give the number of counters moved in each of the wires. 
The manipulations required in the abacus are those re- 
quired in Arabic numerals . What was needed was a 
special symbol for a wire Jn which no counters were 
moved. This was zero, 0 , and Arabic numerals were in 
business , 


T -Formation 


61 



The Granger Collection 


62 


NUMBERS AND COUNTING 


T -Formation 


63 


finding a record-high prime would become trivial and 
could be abandoned. 

However, such a formula has never been found. About 
1600, a French friar named Marin Mersenne proposed a 
formula of partial value which would occasionally, but 
not always, produce a prime. This formula is 2 P — 1 , where 
p is itself a prime number. (You understand, I hope, that 
2 P represents a number formed by multiplying p two's to- 
gether, so that 2 s is 2x2x2x2x2x2x2x2, or 256.) 

Mersenne maintained that the formula would produce 
primes when p was equal to 2, 3, 5, 7, 13, 17, 19, 31, 67, 
127, or 257. This can be tested for the lower numbers 
easily enough. For instance, if p equals 3, then the formula 
becomes 2 3 ~ 1, or 7, which is indeed prime. If p equals 
7, then 2 T — 1 equals 127, which is prime. You can check 
the equation for any of the other values of p you care to. 

The numbers obtained by substituting prime numbers 
for p in Mersenne’s equation are called “Mersenne num- 
bers” and if the number happens to be prime it is a “Mer- 
senne prime.” They are symbolized by the capital letter M 
and the value of p. Thus M 3 equals 7; M T equals 127, and 
so on. 

I don't know what system Mersenne used to decide 
what primes would yield Mersenne primes in his equation, 
but whatever it was, it was wrong. The Mersenne num- 
bers M 2 , M a , M a , M t , M 1S , M 17j M js , M S1 , and M 127 are 
indeed primes, so that Mersenne had put his finger on no 
less than nine Mersenne primes. However, M S7 and M 457 , 
which Mersenne said were primes, proved on painstaking 
examination to be no primes at all. On the other hand, 
M 6J , M 89 , and M 197 , which Mersenne did not list as primes, 
are primes, and this makes a total of twelve Mersenne 
primes. 

In recent years, thanks to computer work, eight more 
Mersenne primes have been located (according to the 
April 1962 issue of Recreational Mathematics ) . These are 
M* ot , M 1279 , M i2D3 , M 22S1 , M 321Tj M 4253 , and M^j*. 
What's more, since that issue, three even larger Mersenne 
primes have been discovered by Donald B, Gillies of the 
University of Illinois. These are M 96Be , M 9941 , and M im *. 

The smallest of these newly discovered Mersenne 
primes, M an , is obtained by working out the formula 
2 a21 ~ 1, You take 521 two's, multiply them together, and 


subtract one. The result is far, far higher than a googol. 
In fact, it is higher than T-13. 

Not to stretch out the suspense, the largest known 
Mersenne prime, M mi ,. and, I believe, the largest prime 
known at present, has 3375 digits and is therefore just 
about T-281%. The googol, in comparison to that, is a 
trifle so small that there is no reasonable way to describe 
its smallness. 

The Greeks played many games with numbers, and one 
of them was to add up the factors of particular integers. 
For instance, the factors of 12 (not counting the number 
itself) are 1, 2, 3, 4, and 6. Each of these numbers, but no 
others, will go evenly into 12. The sum of these factors is 
16, which is greater than the number 12 itself, so that 12 
is an “abundant number.” 

The factors of 10, on the other hand, are 1, 2, and 5, 
which yield a sum of 8. This is less than the number itself, 
so that 10 is a “deficient number.” (All primes are obvi- 
ously badly deficient.) 

But consider 6. Its factors are 1, 2, and 3, and this adds 
up to 6. When the factors add up to the number itself, 
that number is a “perfect number.” 

Nothing has ever come of the perfect numbers in two 
thousand years, but the Greeks were fascinated by them, 
and those of them who were mystically inclined revered 
them. For instance, it could be argued (once Greek cul- 
ture had penetrated Judeo-Christianity) that God had 
created the world in six days because six is a perfect num- 
ber. (Its factors are the first three numbers, and not only 
is their sum six, but their product is also six, and God 
couldn’t be expected to resist all that.) 

I don't know 'whether the mystics also made a point of 
the fact that the lunar month is just a trifle over twenty- 
eight days long, since 28, with factors of 1, 2, 4, 7, and^!4 
(which add up to 28), is another perfect number. Alas, 
the days of the lunar month are actually 29*2 and the 
mystics may have been puzzled over this slipshod arrange- 
ment on the part of the Creator, 

But how many of these wonderful perfect numbers are 
there? Considering that by the time you reach 28, you 
have run into two of them, you might think there were 
many. However, they are rare indeed; far rarer than al- 
most any other well-knowm kind of number. The third 
perfect number is 496, and the fourth is 8128, and 



64 


NUMBERS AND COUNTING 


T-F or motion 


65 


MARIN MERSENNE 

Marin Mersenne war born near the French town of Oize 
on September 8 , 1588. Mersenne war a schoolfellow of 
the great mathematician Rene Descartes. Whereas Des- 
cartes , for some reason , joined the army } for which he 
was singularly ill adapted , Mersenne entered the Churchy 
joining the Minim Friars in 1611. Within the Churchy he 
did yeoman work for science , of which he war an ardent 
exponent . He defended Descartes's philosophy against 
clerical critics, translated some of the works of Galileo . 
and defended him , too. 

Mersenne' s chief service to science was the unusual 
one of serving as a channel for ideas. In the seventeenth 
century , long before scientific journals, international con- 
ferences, and even before the establishment of scientific 
academies , Mersenne was a one-man connecting link 
among the scientists of Fait ope. He wrote voluminous 
letters to regions as distant as Constantinople , informing 
one correspondent of the work of another t making sug- 
gestions arising out of his knowledge of the work of 
many , and constantly urging others to follow this course 
of copious intercommunication. 

He opposed mystical doctrines such as astrology , al- 
chemy, and divination; and strongly supported experimen- 
tation. As a practical example of this belief, he suggested 
to Christiaan Huygens the ingenious notion of timing 
bodies rolling down inclined planes by the use of a pendu- 
lum. This had not occurred to Galileo , who had first 
worked out the principle of the pendulum , but who timed 
his rolling bodies by using water dripping out of a can 
with a hole in the bottom. Huygens took the suggestion 
and it came to fruition in the form of the pendulum clock , 
the first timepiece that was useful to science. 

Mersenne died in Paris on September l f 1648. 



The Granger Collection 


66 


NUMBERS AND COUNTING 


T -Formation 


67 


throughout ancient and medieval times, those were the 
only perfect numbers known. 

The fifth perfect number was not discovered until 
about 1460 (the name of the discoverer is not known) 
and it is 33,550,336. In modern limes, thanks to the help 
of the computer, more and more perfect numbers have 
been discovered and the total now is twenty. The twentieth 
and largest of these is a number with 2663 digits, and this 
is almost equal to T-222. 

But in a way, I have been unfair to Kasner and New- 
man. I have said they invented the googol and I then went 
on to show that it was easy to deal with numbers far 
higher than the googol. However, I should also add they 
invented another number, far, far larger than the googol. 
This second number is the “googolplex,” which is^ defined 
as equal to 10 g<K> * 01 . The exponent, then, is a 1 followed 
by a hundred zeros, and I could write that, but I won t, 
instead, I’ll say that a googolplex can be written as: 

10 10 10 ° or even 10 T ° 10 

The googol itself can be written out easily. I did it at 
the beginning of the article and it only took up a few lines. 
Even the largest number previously mentioned in this 
chapter can be written out with ease. The largest Mer- 
senne prime, if written out in full, would take up less 
than two pages of this book. 

The googolplex, however, cannot be written out— liter- 
ally cannot . It is a 1 followed by a googol zeros, and this 
book will not hold as many as a googol zeros no matter 
how small, within reason, those zeros are printed. In fact, 
you could not write the number on the entire surface of 
the earth, if you made each zero no larger than an atom. 
In fact, if you represented each zero by a nucleon, there 
wouldn’t be enough nucleons in the entire known universe 
or in a trillion like it to supply you with sufficient zeros. 

You can see then that the googolplex is incomparably 
larger than anything I have yet dealt with. And yet I can 
represent it in T-numbers without much trouble. 

Consider! The T-numbers go up through the digits, T-l, 
T-2, T-3, and so on, and eventually reach T- 1,000, 000,- 
000,000. (This is a number equivalent to saying “a trillion 
trillion trillion trillion . . and continuing until you have 
repeated the word trillion a trillion times. It will take you 


umpty-ump lifetimes to do it, but the principle remains.) 
Since we have decided to let a trillion be written as T-l, 
the number T-l, 000, 000, 000, 000 can be written T-(T-l). 

Remember that we must multiply the numerical part 
of the T-number by 12 to get a ten-based exponent. 
Therefore T-(T-l) is equal to iO 12 - 000 - 000 ' 000 '® 00 , which is 
more than 10 1013 

In the same way, we can calculate that T-(T-2) is more 
than 10 loas , and if we continue we finally find that T-(T-8) 
is nearly a googolplex. As for T-(T-9), that is far larger 
than a googolplex; in fact, it is far larger than a googol 
googolplexes. 

One more item and I am through. 

In a book called The Lore of Large Numbers, by Philip 
J. Davis, a number called “Skewes’ number” is given. This 
number was obtained by S. Skewes, a South African 
mathematician who stumbled upon it while working out a 
complex theorem on prime numbers. The number is de- 
scribed as “reputed to be the largest number that has oc- 
curred in a mathematical proof.” It is given as: 

10“ WI 

Since the googolplex is only , Skewes’ number is 

incomparably the greater of the two. 

And how can Skewes’ number be put into T-formation? 

Well, at this point, even I rebel. I’m not going to do it. 

I will leave it to you, O Gentle Reader, and I will tell 
you this much as a hint. It seems to me to be obviously 
greater than T-[T-(T-1)]. 

From there on in, the track is yours and the road to 
madness is unobstructed. Full speed ahead, ail of you. 

As for me, I shall hang back and stay sane; or, at least, 
as sane as I ever am, which isn’t much,* 


[* After this article first appeared , 1 was hounded by readers every now 
and then to write an article on Skewes * number . I finally succumbed and 
an article on this subject , "Skewered!," was written in 1974. You wilt 
find it as the last chapter in my book Of Matters Great and Small 
{Doubleday, 1975)1 



Varieties of the Infinite 


69 


5 VARIETIES 
OF THE 
INFINITE 


There are a number of words that publishers like 
to get into the titles of science-fiction books as an instant 
advertisement to possible fans casually glancing over a dis- 
play that these books are indeed science fiction. Two 
such words are, of course space and time . Others are 
Earth (capitalized), Mars , Venus , Alpha Centauri , tomor- 
row, stars, sun , asteroids, and so on. And one — to get to 
the nub of this chapter — is infinity. 

One of the best s.f. titles ever invented, in my opinion, 
is John Campbell's Invaders from the Infinite. The word 
invaders is redolent of aggression, action, and suspense, 
while infinite brings up the vastness and terror of outer 
space. 

Donald Day’s indispensable Index to the Science Fic- 
tion Magazines lists ‘Infinite Brain,” “Infinite Enemy,” 
“Infinite Eye,” “Infinite Invasion,” “Infinite Moment” 
“Infinite Vision,” and “Infinity Zero” in its title index, and 
I am sure there are many other titles containing the 
word.* 


[* This article first appeared in September 1959 and Donald Day’s Index 
only went up to 1950 . Since 1950 , the popularity of "Infinite" in s.f. 
titles has declined as the literary sophistication of the field has increased . 
Yes, I'm sorry.] 


Yet with all this exposure, with all this familiar use, do 
we know what infinite and infinity mean? Perhaps not all 
of us do. 

We might begin, 1 imagine, by supposing that infinity 
was a large number; a very large number; in fact, the larg- 
est number that could exist. 

If so, that would at once be wrong, for infinity is not a 
large number or any kind of number at all; at least of the 
sort we think of when we say “number.” It certainly isn't 
the largest number that could exist, for there isn't any 
such thing. 

Let's sneak up on infinity by supposing first that you 
wanted to write out instructions to a bright youngster, 
telling him how to go about counting the 538 people who 
had paid to attend a lecture. There would be one particu- 
lar door through which all the audience would leave in 
single file. The youngster need merely apply to each per- 
son one of the various integers in the proper order: 1, 2, 3, 
and so on. 

The phrase “and so on” implies continuing to count until 
all the people have left, and the last person who leaves 
has received the integer 538. If you want to make the 
order explicit, you might tell the boy to count in the fol- 
lowing fashion and then painstakingly list all the integers 
from 1 to 538. This would undoubtedly be unbearably 
tedious, but the boy you are dealing with is bright and 
knows the meaning of a gap containing a dotted line, so 
you write: “Count thus: 1, 2, 3, ... , 536, 537, 538.” 
The boy will then understand (or should understand) that 
the dotted line indicates a gap to be filled by all the in- 
tegers from 4 to 535 inclusive, in order and without 
omission. 

Suppose you didn’t know what the number of the audi- 
ence was. It might be 538 or 427 or 651. You could in- 
struct the boy to count until an integer had been given to 
the last man, whatever the man, whatever the integer. To 
express that symbolically, you could write thus: “Count: 
1, 2, 3, ... , n — 2, 7i — 1, n.” The bright boy would un- 
derstand that n routinely represents some unknown but 
definite integer. 

Now suppose the next task you set your bright young- 
ster was to count the number of men entering a door, 
filing through a room, out a second door, around the 


68 



70 


NUMBERS AND COUNTING 


LARGE NUMBERS 

The fact is that in ancient times there was little need for 
large numbers . The largest number-name used was gen- 
erally "thousand.” If larger numbers were needed, 
phrases were used (as by us) of tens of thousands, and of 
hundreds of thousands. In ancient times, one went beyond 
and spoke of thousands of thousands. The word “ million ” 
(from an Italian word meaning " large thousand ") for a 
thousand-thousand was only invented in the late Middle 
Ages when commerce had revived to the point where 
thousand-thousands were common enough in bookkeep- 
ing to make a special word convenient. (Billions, trillions, 
etc., followed later , and to this day the use of the larger 
numbers is unsettled. In the United States, a billion is a 
thousand million; in Great Britain, a billion is a million 
million.) 

We can see the ancient poverty of names for numbers 
if we read the Bible. The largest number specifically 
named in the Bible occurs in 2 Chronicles 14:9 where a 
battle is described between Ethiopian invaders and the 
forces of Asa, King of Judah , "And there came out 
against them Zerah the Ethiopian with an host of a 
thousand thousand. . . /' Grossly exaggerated, of course, 
but the only mention of a number as high as a million 
in the Bible. 

Elsewhere, when the need for large numbers is re- 
quired, only comparisons can be made . Thus, in Genesis 
22:17 , God promises Abraham (who had just shown him- 
self to be willing to sacrifice his only son to God), " / wilt 
multiply thy seed as the stars of the heaven and as the 
sand which is upon the sea shore ” ( That almost seems to 
apply to the illustration, which shows a crowd in Rio de 
Janeiro protesting the sinking of neutral Brazilian ships 
shortly before Brazil declared war on Germany and 
Italy.) 

There was even the feeling that there are numbers so 
enormous that they can’t be counted : Thus Solomon 
speaks of his subjects as “a great people , that cannot be 
numbered nor counted for multitude ” (1 Kings 3:8). In 
the third century b.c., Archimedes demonstrated, for the 
first time, that any finite quantity can be numbered easily. 



The Granger Collection 


72 


NUMBERS AND COUNTING 


Varieties of the Infinite 


73 


building, and through the first door again, the men form- 
ing a continuous closed system. 

Imagine both marching men and counting boy to be 
completely tireless and willing to spend an eternity in their 
activities. Obviously the task would be endless. There 
would be no last man at all, ever, and there is no last 
integer at all, ever. (Any integer, however large, even if it 
consisted of a series of digits stretching in microscopic 
size from here to the farthest star, can easily be increased 
by 1.) 

How do we write instructions for the precise counting 
involved in such a task. We can write: “Count thus: 1, 2, 
3, and so on endlessly. 11 

The phrase “and so on endlessly 11 can be written in 
shorthand, thus, x . 

The statement “1, 2, 3, . . . , x” should be read “one, 
two, three, and so on endlessly” or “one, two, three, and 
so on without limit,” but it is usually read, “one, two, 
three, and so on to infinity.” Even mathematicians intro- 
duce infinity here, and George Gamow, for instance, has 
written a most entertaining book entitled just that: One, 
Two , Three . . . Infinity. 

It might seem that using the word infinity is all right, 
since it comes from a Latin word meaning “endless,” but 
nevertheless it would be better if the Anglo-Saxon were 
used in this case. The phrase “and so on endlessly” can't 
be mistaken. Its meaning is clear. The phrase “and so on 
to infinity,” on the other hand, inevitably gives rise to the 
notion that infinity is some definite, though very huge, in- 
teger and that once we reach it we can stop. 

So let’s be blunt. Infinity is not an integer or any num- 
ber of a kind with which we are familiar. It is a quality; 
a quality of endlessness. And any set of objects (numbers 
or otherwise) that is endless can be spoken of as an “in- 
finite series” or an “infinite set.” The list of integers from 
I on upward is an example of an “infinite set.” 

Even though x is not a number, we can still put it 
through certain arithmetical operations. We can do that 
much for any symbol. We can do it for letters in algebra 
and write a-\-b~c. Or we can do it for chemical formulas 
and write: CH 4 -i-30 i =C0 £ H-2H a 0* Or we can do it for 
abstractions, such as: Man + Woman = Trouble. 

The only thing we must remember is that in putting 


symbols that are not integers through arithmetical paces, 
we ought not to be surprised if they don't follow the ordi- 
nary rules of arithmetic which, after all, were originally 
worked out to apply soecificallv to integers. 

For instance, 3-2=1, 17-2 = 15, 4875-2=4873. In 
general, any integer, once 2 is subtracted, becomes a dif- 
ferent integer. Anything else is unthinkable. 

But now suppose we subtract 2 from the unending series 
of integers. For convenience sake, we can omit the first 
two integers, 1 and 2, and start the series: 3, 4, 5, and so 
on endlessly. You see, don’t you, that you can be just as 
endless starting the integers at 3 as at 1. so that you can 
write: 3, 4, 5, ... , x. 

In other words, when two items are subtracted from an 
infinite set, what remains is still an infinite set. In symbols, 
we can write this: x — 2~ x. This looks odd because we 
are used to integers, where subtracting 2 makes a differ- 
ence. But infinity is not an integer and works by different 
rules. (This can’t be repeated often enough.) 

For that matter, if you lop off the first 3 integers or 
the first 25 or the first 1000000000000, what is left of the 
series of integers is still endless. You can always start, 
say, with 1000000000001, 1000000000002, and go on 
endlessly. So ^—n=o o, where n represents any integer, 
however great. 

In fact, we can be more startling than that. Suppose we 
consider only the even integers. We would have a series 
that would go: 2, 4, 6, and so on endlessly. It would be 
an infinite series and could therefore be written: 2, 4, 6, 
. . . , oc. In the same way, the odd integers would form 
an infinite series and could be written: 1, 3, 5, . . . , oo. 

Now, then, suppose you went through the series of 
integers and crossed out every even integer you came to, 
thus: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, . . . , From 
the infinite series of integers, you would have eliminated 
an infinite series of even integers and you would have left 
behind an infinite series of odd integers. This can be 
symbolized as x ~ x = x . 

Furthermore, it could work the other way about. If you 
started with the even integers only and added one odd 
integer, or two, or five; or a trillion, you would still merely 
have an unending series, so that x +«= x. In fact, if you 
added the unending series of odd integers to the unending 



74 


NUMBERS AND COUNTING 


Varieties of the Infinite 


75 


series of even integers, you would simply have the unend* 
ing series of all integers, or : x -f x ~ x . 

By this point, however, it is just possible that some of 
you may suspect me of pulling a fast one. 

After all, in the first 10 integers, there are 5 even in- 
tegers and 5 odd ones; in the first 1000 integers, there are 
500 even integers and 500 odd integers; and so on. No 
matter how T many consecutive integers we take, half are 
always even and half are odd. 

Therefore, although the series 2, 4, 6, . . . is endless, 
the total can only be half as great as the total of the also 
endless series 1, 2, 3, 4, 5, 6. . . . And the same is true 
for the series 1, 3, 5, , which, though endless, is only 

half as great as the series of all integers. 

And so (you might think) in subtracting the set of even 
integers from the set of all integers to obtain the set of 
odd integers, what we are doing can be represented as: 
oo^ : !4go=%x. That, you might think with a certain sat- 
isfaction, ‘‘makes sense/’ 

To answer that objection, let's go back to counting the 
unknown audience at the lecture. Our bright boy, who 
has been doing all our counting, and is tired of it, turns 
to you and asks, “How many seats are there in the lecture 
hall? 1 ' You answer, “640/’ 

He thinks a little and says, “Well, I see that every seat 
is taken. There are no empty seats and there is no one 
standing.” 

You, having equally good eyesight, say, “That’s right.” 

“Well, then,” says the boy, “why count them as they 
leave. We know right now that there are exactly 640 spec- 
tators.” 

And he's correct. If two series of objects (A series and 
B series) just match up so that there is one and only one 
A for every B and one and only one B for every A, then 
we know that the total number of A objects is just equal 
to the total number of B objects. 

In fact, this is what we do when we count. If we want 
to know how many teeth there are in the fully equipped 
human mouth, we assign to each tooth one and only one 
number (in order) and we apply each number to one and 
only one tooth. (This is called placing two series into 
“one-to-one correspondence.”) We find that we need only 
32 numbers to do this, so that the series 1, 2, 3, ... , 


30, 31, 32 can be exactly matched with the series one 
tooth, next tooth, next tooth, . . . , next tooth, next tooth, 
last tooth. 

And therefore, we say, the number of teeth in the fully- 
equipped human mouth is the same as the number of in- 
tegers from 1 to 32 inclusive. Or, to put it tersely and 
succinctly: there are 32 teeth. 

Now wc can do the same for the set of even integers. 
We can write down the even integers and give each one a 
number. Of course, we can't write down all the even in- 
tegers, but we can write down some and get started any- 
way. We can write the number assigned to each even 
integer directly above it, with a double-headed arrow, so: 

12345 6 7 8 9 10... 

t t t t t t : t t x 

2 4 6 8 10 12 14 36 18 20 .. . 

We can already see a system here. Every even integer 
is assigned one particular number and no other, and you 
can tell what the particular number is by dividing the even 
integer by 2. Thus, the even integer 38 has the number 19 
assigned to it and no other. The even integer 24618 has 
the number 12309 assigned to it. In the same way, any 
given number in the series of all integers can be assigned 
to one and only one even integer. The number 538 is ap- 
plied to even integer 1076 and to no other. The number 
29999999 is applied to even integer 59999998 and no 
other; and so on. 

Since every number in the series of even integers can 
be applied to one and only one number in the series of all 
integers and vice versa, the two series are in one-to-one 
correspondence and are equal. The number of even in- 
tegers then is equal to the number of all integers. By a 
similar argument, the number of odd integers is equal to 
the number of all integers. 

You may object by saying that when all the even in- 
tegers (or odd integers) are used up, there will still be fully 
half the series of all integers left over. Maybe so, but this 
argument has no meaning since the series of even integers 
(or odd integers) will never be used up. 

Therefore, when we say that “all integers” minus “even 
integers” equals “odd integers,” this is like saying x — x 
— x, and terms like can be thrown out. 



76 


NUMBERS AND COUNTING 


Varieties of the Infinite 


77 


In fact, in subtracting even integers from all integers, 
we are crossing out every other number and thus, in a way, 
dividing the series by 2. Since the series is still unending, 
oc/2— oo anyway, so what price half of infinity? 

Better yet, if we crossed out every other integer in the 
series of even integers, we would have an unending series 
of integers divisible by 4; and if we crossed out every other 
integer in that series, we would have an unending series of 
integers divisible by 8, and so on endlessly. Each one of 
these “smaller’' series could be matched up with the series 
of all integers in one-to-one correspondence. If an unend- 
ing series of integers can be divided by 2 endlessly, and 
still remain endless, then we are saying that oc / oo — oo . 

If you doubt that endless series that have been drasti- 
cally thinned out can be put into one-to-one correspond- 
ence with the series of all integers, just consider those 
integers that are multiples of one trillion. You have: 
1 ,000,000,000,000; 2,000,000,000,000; 3,000,000,000,000; 
♦ . . ; oo. These are matched up with 1, 2, 3, . . . , x. 
For any number in the set of “trillion-integers,” say 
4,856,000,000,000,000, there is one and only one number 
in the set of all integers, which, in this case, is 4856. For 
any number in the set of all integers, say 342, there is one 
and only one number in the set of "trillion-integers,” in 
this case, 342,000,000,000,000. Therefore, there are as 
many integers divisible by a trillion as there are integers 
altogether. 

It works the other way around, too. If you place be- 
tween each number the midway fraction, thus: 1, V-A t 

2, 2 Yz, 3, 3Yz, , . . , oo, you are, in effect, doubling the 
number of items in the series and yet this new series can 
be put into one-to-one correspondence with the set of in- 
tegers, so that 2 oo — oo. In fact, if you keep on doing it 
indefinitely, putting in all the fourths, then all the eighths, 
then all the sixteenths, you can still keep the resulting 
series in one-to-one correspondence with the set of all 
integers so that oc - oo — x 2 = oo - 

This may seem too much to swallow. How can all the 
fractions be lined up so that we can be sure that each one 
is getting one and only one number. It is easy to line up 
integers, 1, 2, 3, or even integers, 2, 4, 6, or even prime 
numbers 2, 3, 5, 7, II. . . . But how can you line up frac- 


tions and be sure that all are included, even fancy ones 
like and 

There are, however, several ways to make up an inclusive 
list of fractions. Suppose we first list all the fractions in 
which the numerator and denominator add up to 2. There 
is only one of these: Yu Then list those fractions where 
the numerator and denominator add up to 3. There are 
two of these: ?i and ¥>. Then we have Yu ti , and where 
the numerator and denominator add up to 4. Then we 
have Yu % % and 14. In each group, you see, we place 
the fractions in the order of decreasing numerator and 
increasing denominator. 


If we make such a list: 


w, % 


71 , 


l/> 


Yu %, 


%, ri. Y, and so on endlessly we can be assured that 
any particular fraction, no matter how complicated, will 
be included if we proceed far enough. The fraction 
i480fi ^ 25^3 will be in that group of fractions in which the 
numerator and denominator add up to 2740422, and it 
will be the 2725523rd of the group. Similarly, 68944447 ^ 
will be the second fraction in the group in which the 
numerator and the denominator add up to 689444475. 
Every possible fraction will thus have its particular as- 
signed place in the series. 

It follows, then, that every fraction has its own number 
and that no fraction will be left out. Moreover, every num- 
ber has its own fraction and no number is left out. The 
series of all fractions is put into a one-to-one correspond- 
ence with the scries of all integers, and thus the number 
of all fractions is equal to the number of all integers. 

(In the list of fractions above, you will see that some 
are equal in value. Thus, Y and % are listed as different 
fractions, but both have the same value. Fractions like M, 
Yz, and % not only have the same value but that value is 
that of an integer, 1. All this is all right. It shows that the 
total number of fractions is equal to the total number of 
integers even though in the series of fractions, the value of 
each particular fraction, and all integral values as well, is 
repeated many times; in fact, endlessly.) 


By now you may have more or less reluctantly decided 
that all unendingness is the same unendingness and that 
“infinity” is “infinity” no matter what you do to it. 

Not so! 

Consider the points in a line. A line can be marked off 



78 


NUMBERS AND COUNTING 


Varieties of the Infinite 


79 


at equal intervals, and the marks can represent points 
which are numbered 1, 2, 3, and so on endlessly, if you 
imagine the line continuing endlessly. The midpoints be- 
tween the integer-points can be marked V 2 , Vh, , . . , 

and when the thirds can be marked and the fourths and 
the fifths and indeed all the unending number of fractions 
can be assigned to some particular point. 

It would seem then that every point in the line would 
have some fraction or other assigned to it. Surely there 
would be no point in the line left out after an unending 
number of fractions had been assigned to it? 

Oh, wouldn’t there? 

There is a point on the line, you see, that would be 
represented by a value equal to the square root of two 
(y 2). This can be shown as follows. If you construct 
a square on the line with each side exactly equal to the 
interval of one integer already marked off on the line, then 
the diagonal of the square would be just equal to V 2. If 
that diagonal is laid down on the line, starting from the 
zero point, the end of that diagonal coincid es with the 
point on the line which can be set equal to V 2. 

Now the catch is that the value of y 2 cannot be rep- 
resented by a fraction; by any fraction; by any conceivable 
fraction. This was proved by the ancient Greeks and the 
proof is simple but I’ll ask you to take my word for it here 
to save room. Well, if all the fractions are assigned to 
various points in the line, at least one point, that which 
corresponds to \/ 27 will be left out. 

All numbers which can be represented as fractions are 
“rational numbers 11 because a fraction is really the ratio 
of two numbers, the numerator and the denominator. 
Numbers which cannot be rep resented as fractions are 
“irrational numbers 11 and \/2~ is no means the . onl V 
one of those, although it was the first such to be discov- 
ered. Most square roots, cube roots, fourth roots, etc., are 
irrationals, so are most sines, cosines, tangents, etc., so are 
numbers involving pi (tt), so are logarithms. 

In fact, the set of irrational numbers is unending. It can 
be shown that between any two points represented by ra- 
tional numbers on a line, however close those two points 
are, there is always at least one point represented by an 
irrational number. 

Together, the rational numbers and irrational numbers 
are spoken of as “real numbers.” It can be shown that any 


given real number can be made to correspond to one and 
only one point in a given line; and that any point in the 
line can be made to correspond to one and only one real 
number. In other words, a point in a line which can’t be 
assigned a fraction, can always be assigned an irrational. 
No point can be missed by both categories. 

The series of real numbers and the series of points in a 
line are therefore in one-to-one correspondence and are 
equal. 

Now the next question is: Can the series of all real 
numbers, or of all points in a line (the two being equiva- 
lent), be set into a one-to-one correspondence with the 
series of integers. The answer is, No f 

It can be shown that no matter how you arrange your 
real numbers or your points, no matter what conceivable 
system you use, an endless number of either real numbers 
or points will always be left out. The result is that we are 
in the same situation as that in which we are faced with 
an audience in which all seats are taken and there are 
people standing. We are forced to conclude that there are 
more people than seats. And so, in the same way, we 
are forced to conclude that there are more real numbers, 
or points in a line, than there are integers. 

If we want to express the endless series of points by 
symbols, we don’t want to use the symbol x for “and so 
on endlessly,” since this has been all tied up with integers 
and rational numbers generally. Instead, the symbol C is 
usually used, standing for continuum , since all the points 
in a line represent a continuous line. 

We can therefore write the series: Point 1, Point 2, 
Point 3, . , . , C. 

Now we have a variety of endlessness that is different 
and more intensely endless than the endlessness repre- 
sented by “ordinary infinity.” 

This new and more intense endlessness also has its 
peculiar arithmetic. For instance, the points in a short line 
can be matched up one-for-one with the points in a long 
line, or the points in a plane, or the points in a solid. In 
fact, let’s not prolong the agony, and say at once that there 
are as many points in a line a millionth of an inch long as 
there are points in all of space. 

About 1895 the German mathematician Georg Cantor 
Worked out the arithmetic of infinity and also set up a 



80 


NUMBERS AND COUNTING 


Varieties of the Infinite 


SI 


GEORG CANTOR 

To designate Cantor by nationality is difficult. He was 
bom in Russia, in Leningrad, in fact (it was called St. 
Petersburg then), on March 3 , 1845, His father had emi- 
grated to Russia from Denmark , however, and then left 
Russia for Germany when young Georg hyw only eleven. 
In addition, the family was of Jewish descent, though his 
mother wav born a Roman Catholic and his father was 
converted to Protestantism , 

Even as a schoolboy Cantor showed talent for mathe- 
matics , and eventually (over his father's objections) he 
made mathematics his profession. In 1867 he obtained his 
Ph.D. magna cum laude from the University of Berlin. He 
obtained an academic position at the University of Halle, 
advancing to a professorial appointment in 1872. 

It was in 1874 that Cantor began to introduce his 
intellect-shaking concepts of infinity. Earlier, Galileo had 
caught glimpses of it, but Cantor was the first to erect a 
complete logical structure in which a whole series of 
transfinite numbers was postulated , representing different 
orders of infinity r so to speak . 

Not much can be done with the different orders in 
terms of sets that can be described . The set of integers 
is equal to the first , the set of real numbers is higher , the 
set of functions is higher still , and there we must stop. 

Cantor* s views were not accepted by all his colleagues. 
In particular Leopold Kronecker , who had been one of 
Cantor's teachers, attacked Cantor's work with great vigor . 
Inspired by professional jealousy, Kronecker prevented 
Cantor's advancement t keeping him from a post at the 
University of Berlin. Cantor's mental health broke in 
1884 under the strains of the controversy , and he died in 
a mental hospital in Halle, Saxony, on January 6, 1918 . 



The Granger Collection 



82 


NUMBERS AND COUNTING 


Varieties of the Infinite 


83 


whole series of different varieties of endlessnesses, which 
he called “transfinite numbers.” 

He represented these transfinite numbers by the letter 
aleph, which is the first letter of the Hebrew alphabet and 
which looks like this: ^ 

The various transfinities can be listed in increasing size 
or, rather, in increasing intensity of endlessness by giving 
each one a subscript, beginning with zero. The very lowest 
transfinite would be “aleph-null,” then there w'ould be 
“aleph-one,” 6 'aleph -two,” and so on, endlessly. 

This could be symbolized as: Hi* ■ • ■ » 

Generally, whatever you do to a particular transfinite 
number in the way of adding, subtracting, multiplying, or 
dividing, leaves it unchanged. A change comes only when 
you raise a transfinite to a transfinite power equal to itself 
(not to a transfinite power less than itself). Then it is in- 
creased to the next higher transfinite. Thus: 

and so on. 

What we usually consider as infinity, the endlessness of 
the integers, has been shown to be equal to aleph-null. In 
other words: cc — Xo- And so the tremendous vastness of 
ordinary infinity turns out to be the very smallest of all 
the transfinites. 

That variety of endlessness which we have symbolized as 
C may be represented by aleph-one so that C = Wi> but 
this has not been proved. No mathematician has yet been 
able to prove that there is any infinite series which has an 
endlessness more intense than the endlessness of the in- 
tegers but less intense than the endlessness of the points in 
a line. However, neither has any mathematician been able 
to prove that such an intermediate endlessness does not 
exist.* 

If the continuum is equal to aleph-one, then w f e can 
finally write an equation for our friend “ordinary infinity 
which will change it: 


the curves that can be drawn on a plane is even more 
intense than the endlessness of points in a line. In other 
words, there is no way of lining up the curves so that they 
can be matched one-to one with the points in a line, with- 
out leaving out an unending series of the curves. This end- 
lessness of curves may be equal to aleph-two, but that 
hasn’t been proved yet, either. 

And that is all. Assuming that the endlessness of inte- 
gers is aleph-null, and the endlessness of points is aleph- 
one, and the endlessness of curves is aleph-two, we have 
come to the end. Nobody has ever suggested any variety 
of endlessness which could correspond to aleph-three (let 
alone to aleph-thirty or aleph-three-million) . 

As John E, Freund says in his book A Modern Intro- 
duction to Mathematics * (a book I recommend to all 
who found this article in the least interesting), “It seems 
that our imagination does not permit us to count beyond 
three when dealing with infinite set s.” 

Still, if we now return to the title Invaders from the 
Infinite, I think we are entitled to ask, with an air of 
phlegmatic calm, “Which infinite? Just aleph-null? Nothing 
more?” 


* New York: Prentice-Hall, 1956. 


co * — C- 

Finally, it has been shown that the endlessness of all 


[ 4 Since this article first appeared f it has been shown that the statement 
C— Aleph-one can be neither proved nor disproved by any method .] 



Part II 


NUMBERS AND 
MATHEMATICS 




6 a piece 

OF PI 


In my essay “Those Crazy Ideas,” which appeared 
in my book Fact and Fancy (Doubleday, 1962), I casually 
threw in a footnote to the effect that — 1 . Behold, a 
good proportion of the comment which I received there- 
after dealt not with the essay itself but with that footnote 
(one reader, more in sorrow than in anger, proved the 
equality, which I had neglected to do). 

My conclusion is that some readers are interested in 
these odd symbols. Since I am, too (albeit I am not really 
a mathematician, or anything else), the impulse is irre- 
sistible to pick up one of them, say x, and talk about it in 
this chapter and the next. In Chapter 8, I will discuss i. 

In the first place, what is 7 r ? Well, it is the Greek letter 
pi and it represents the ratio of the length of the perimeter 
of a circle to the length of its diameter, Perimeter is from 
the Greek perimetron, meaning “the measurement 
around,” and diameter from the Greek diametron t mean- 
ing “the measurement through.” For some obscure reason, 
while it is customary to use perimeter in the case of poly- 
gons, it is also customary to switch to the Latin circum- 
ference in speaking of circles. That is all right, I suppose 
(I am no purist), but it obscures the reason for the sym- 
bol t r- 

Back about 1600 the English mathematician William 
87 



88 


NUMBERS AND MATHEMATICS 


A Piece of Pi 


89 


Oughtred, in discussing the ratio of a circle’s perimeter 
to its diameter, used the Greek letter x to symbolize the 
perimeter and the Greek letter 8 (delta)- to symbolize the 
diameter. They were the first letters, respectively of 
perimetron and diametron. 

Now mathematicians often simplify matters by setting 
values equal to unity whenever they can. For instance, 
they might talk of a circle of unit diameter. In such a cir- 
cle, the length of the perimeter is numerically equal to the 
ratio of perimeter to diameter. (This is obvious to some 
of you. I suppose, and the rest of you can take my word 
for it.) Since in a circle of unit diameter the perimeter 
equals the ratio, the ratio can be symbolized by *■, the 
symbol of the perimeter. And since circles of unit diame- 
ter are very frequently dealt with, the habit becomes 
quickly ingrained. 

The first top-flight man to use ?r as the symbol for the 
ratio of the length of a circle’s perimeter to the length of 
its diameter was the Swiss mathematician Leonhard Euler, 
in 1737, and what was good enough for Euler was good 
enough for everyone else. 

Now I can go back to calling the distance around a 
circle the circumference. 

But what is the ratio of the circumference of a circle 
to its diameter in actual numbers? 

This apparently is a question that always concerned the 
ancients even long before pure mathematics was invented. 
In any kind of construction past the hen-coop stage you 
must calculate in advance all sorts of measurements, if 
you are not perpetually to be calling out to some under- 
ling, “You nut, these beams are all half a foot too short.” 
In order to make the measurements, the universe being 
what it is, you are forever having to use the value of tt in 
multiplications. Even when you’re not dealing with circles, 
but only with angles (and you can’t avoid angles) you 
will bump into ? r. 

Presumably, the first empirical calculators who realized 
that the ratio was important determined the ratio by draw- 
ing a circle and actually measuring the length of the di- 
ameter and the circumference. Of course, measuring the 
length of the circumference is a tricky problem that can’t 
be handled by the usual wooden foot-rule, which is far 
too inflexible for the purpose. 

What the pyramid-builders and their predecessors prob- 


ably did was to lay a linen cord along the circumference 
very carefully, make a little mark at the point where the 
circumference was completed, then straighten the line and 
measure it with the equivalent of a wooden foot-rule. 
(Modern theoretical mathematicians frown at this and 
make haughty remarks such as “But you are making the 
unwarranted assumption that the line is the same length 
when it is straight as when it was curved.” I imagine the 
honest workman organizing the construction of the local 
temple, faced with such an objection, would have solved 
matters by throwing the objector into the river Nile.) 

Anyway, by drawing circles of different size and mak- 
ing enough measurements, it undoubtedly dawned upon 
architects and artisans, very early in the game, that the 
ratio was always the same in all circles. In other words, 
if one circle had a diameter twice as long or 1 % as long 
as the diameter of a second, it would also have a circum- 
ference twice as long or 1% as long. The problem boiled 
down, then, to finding not the ratio of the particular circle 
you were interested in using, but a universal ratio that 
would hold for all circles for all time. Once someone had 
the value of ^ in his head, he would never have to deter- 
mine the ratio again for any circle. 

As to the actual value of the ratio, as determined by 
measurement, that depended, in ancient times, on the care 
taken by the person making the measurement and on the 
value he placed on accuracy in the abstract. The ancient 
Hebrews, for instance, were not much in the way of con- 
struction engineers, and when the time came for them to 
build their one important building (Solomon’s temple), 
they had to call in a Phoenician architect. 

It is to be expected, then, that the Hebrews in describ- 
ing the temple would use round figures only, seeing no 
point in stupid and troublesome fractions, and refusing to 
be bothered with such petty and niggling matters when 
the House of God was in question. 

Thus, in Chapter 4 of 2 Chronicles, they describe a 
“molten sea” which was included in the temple and which 
was, presumably, some sort of container in circular form. 
The beginning of the description is in the second verse of 
that chapter and reads: “Also he made a molten sea of ten 
cubits from brim to brim, round in compass, and five 
cubits the height thereof; and a line of thirty cubits did 
compass it round about.” 



90 


NUMBERS AND MATHEMATICS 


ARCHIMEDES 

Archimedes , the son of an astronomer , was the greatest 
scientist and mathematician of ancient times , and his equal 
did not arise until Isaac Newton, two thousand years 
later , Although educated in the great university city of 
Alexandria , he did his work in his native city of Syracuse 
in Sicily, where he had been born about 287 b.c. He 
seems to have been a relative of the Syracusan King , 
Hieron II, and wealthy enough to carry on his work at 
leisure . 

Archimedes worked out the principle of the lever , and 
also the principle of buoyancy, which made it possible to 
tell that a gold crown had been adulterated with copper 
without destroying the crown , Archimedes saw the prin- 
ciple in a flash while in the bath , and it was on that 
occasion that he went running through Syracuse in the 
nude shouting “Eureka, Eureka!*’ f " I have iff I have it!’*). 

The most fascinating tales about him come toward the 
end of his long life , when Syracuse deserted its alliance 
with the Roman Republic and when , in consequence, a 
Roman fleet laid siege to the city. Archimedes was a one- 
man defense at the time, thinking up ingenious devices 
to do damage to the fleet . He is supposed to have con- 
structed large lenses to set fire to the fleet , mechanical 
cranes to lift the ships and turn them upside down, and 
so on. In the end t the story goes t the Romans dared not 
approach the walls too closely and would flee if as much 
as a rope showed above it. 

In 212 B.c the city was taken, however , after a three- 
year siege. The Roman commander ordered that Archi- 
medes be taken alive, but he was engaged in a mathe- 
matical problem at the time, and when a soldier ordered 
him to come along, he refused to leave his figures in the 
sand , The soldier killed him . 


\ I 



Culver Pictures, !nc . 


92 


NUMBERS AND MATHEMATICS 


A Piece of Pi 


93 


The Hebrews, you see, did not realize that in giving the 
diameter of a circle (as ten cubits or as anything else) 
they automatically gave the circumference as well. They 
felt it necessary to specify the circumference as thirty 
cubits and in so doing revealed the fact that they con- 
sidered tt to be equal to exactly 3. 

There is always the danger that some individuals, too 
wedded to the literal words of the Bible, may consider 3 
to be the divinely ordained value of x in consequence . I 
wonder of this may not have been the motive of the sim- 
ple soul in some state legislature who some years back, 
introduced a bill which would have made x legally equal 
to 3 inside the bounds of the state. Fortunately, the bill 
did not pass or all the wheels in that state (which would, 
of course, have respected the laws of the state's august 
legislators) would have turned hexagonal. 

In any case, those ancients who were architecturally 
sophisticated knew well, from their measurements, that 
the value of 7 r was distinctly more than 3. The best value 
they had was 2 % (or 3#, if you prefer) which really isn’t 
bad and is still used to this day for quick approximations. 

Decimally, —h is equal, roughly, to 3.142857 . , . , 
while x is equal, roughly, to 3.141592. . . . Thus, 22 h is 
high by only 0.04 per cent or I part in 2500. Good enough 
for most rule-of-thumb purposes. 

Then along came the Greeks and developed a system of 
geometry that would have none of this vile lay-down-a- 
string-and-measure-it-with-a-ruler business. That, obvi- 
ously, gave values that were only as good as the ruler and 
the string and the human eye, all of which were dreadfully 
imperfect. Instead, the Greeks went about deducing what 
the value of x must be once the perfect lines and curves 
of the ideal plane geometry they had invented were taken 
properly into account. 

Archimedes of Syracuse, for instance, used the “method 
of exhaustion” (a forerunner of integral calculus, which 
Archimedes might have invented two thousand years be- 
fore Newton if some kind benefactor of later centuries 
had only sent him the Arabic numerals via a time ma- 
chine) to calculate x. 

To get the idea, imagine an equilateral triangle with its 
vertexes on the circumference of a circle of unit diameter. 
Ordinary geometry suffices to calculate exactly the perime- 


ter of that triangle. It comes out to if you are 

curious, or 2.598076. . . . This perimeter has to be less 
than that of the circle (that is, than the value of x), again 
by elementary geometrical reasoning. 

Next, imagine the arcs between the vertexes of the tri- 
angle divided in two so that a regular hexagon (a six-sided 
figure) can be inscribed in the circle. Its perimeter can be 
determined also (it is exactly 3) and this can be shown to 
be larger than that of the triangle but still less than that of 
the circle. By proceeding to do this over and over again, 
a regular polygon with 12, 24, 48 . . . sides can be 
inscribed. 

The space between the polygon and the boundary of the 
circle is steadily decreased or “exhausted” and the polygon 
approaches as close to the circle as you wish, though it 
never really reaches it. You can do the same with a series 
of equilateral polygons that circumscribe the circle (that 
lie outside it, that is, with their sides tangent to the circle) 
and get a series of decreasing values that approach the 
circumference of the circle. 

In essence, Archimedes trapped the circumference be- 
tween a series of numbers that approached x from below, 
and another that approached it from above. In this way x 
could be determined with any degree of exactness, pro- 
vided you were patient enough to bear the tedium of 
working with polygons of large numbers of sides. 

Archimedes found the time and patience to work with 
polygons of ninety-six sides and was able to show that the 
value of x was a little below and a little above the 
slightly smaller fraction 2 %. 

Now the average of these two fractions is 312 %94 and 
the decimal equivalent of that is 3.141851. . . , This is 
more than the true value of x by only 0.0082 per cent or 
1 part in 12,500. 

Nothing better than this was obtained, in Europe, at 
least, until the sixteenth century. It was then that the frac- 
tion 35 %i3 was first used as an approximation of x- This 
is really the best approximation of x that can be expressed 
as a reasonably simple fraction. The decimal value of 
35 %i3 is 3.14159292 . . . , while the true value of x is 
3.14159265. . . , You can see from that that 35 %i3 is 
higher than the true value by only 0.000008 per cent, or 
by one part in 12,500,000. 



94 


NUMBERS AND MATHEMATICS 


A Piece of Pi 


95 


Just to give you an idea of how good an approximation 
355 Ai 3 is, let’s suppose that the earth were a perfect sphere 
with a diameter of exactly 8000 miles. We could then 
calculate the length of the equator by multiplying 8000 
by t T . Using the approximation 35 %i 3 for x, the answer 
comes out 25,132.7433 . , . miles. The true value of ^ 
would give the answer 25,132.7412 . . . miles. The differ- 
ence would come to about 1 1 feet. A difference of 1 1 feet 
in calculating the circumference of the earth might well 
be reckoned as negligible. Even the artificial satellites that 
have brought our geography to new heights of precision 
haven’t supplied us with measurements within that range 
of accuracy. 

It follows then that for anyone but mathematicians, 
35 %i3 is as close to tt as it is necessary to get under any 
but the most unusual circumstances. And yet mathema- 
ticians have their own point of view. They can’t be happy 
without the true value. As far as they are concerned, a 
miss, however close, is as bad as a megaparsec. 

The key step toward the true value was taken by 
Francois Vieta, a French mathematician of the sixteenth 
century. He is considered the father of algebra because, 
among other things, he introduced the use of letter sym- 
bols for unknowns, the famous x’s and /s, which most of 
us have had, at one time or another in our lives, to face 
with trepidation and uncertainty. 

Vieta performed the algebraic equivalent of Archimedes’ 
geometric method of exhaustion. That is, instead of setting 
up an infinite series of polygons that came closer and 
closer to a circle, he deduced an infinite series of fractions 
which could be evaluated to give a figure for tt. The greater 
the number of terms used in the evaluation, the closer you 
were to the true value of tt. 

I won’t give you Vieta’s series here because it involves 
square roots and the square roots of square roots and the 
square roots of square roots of square roots. There is no 
point in involving one’s self in that when other mathema- 
ticians derived other series of terms (always an infinite 
series) for the evaluation of tt\ series much easier to write. 

For instance, in 1673 the German mathematician Gott- 
fried Wilhelm von Leibniz (who first worked out the 


binary system — see Chapter 2) derived a series which can 
be expressed as follows: 

* = %-% + %■ + + - . - 

Being a naive nonmathematician myself, with virtually 
no mathematical insight worth mentioning, I thought, 
when I first decided to write this essay, that I would use 
the Leibniz series to dash off a short calculation and show 
you how it would give x easily to a dozen places or so. 
However, shortly after beginning. I quit. 

You may scorn my lack of perseverance, but any of you 
are welcome to evaluate the Leibniz series just as far as 
it is written above, to 4 /i 5 , that is. You can even drop me a 
postcard and tell me the result, If, when you finish, you 
are disappointed to find that your answer isn’t as close to 
x as the value of 35 *?ii 3 , don’t give up. Just add more terms. 
Add *i 7 to your answer, then subtract *4n, then add %i 
and subtract %z, and so on. You can go on as long as you 
want to, and if any of you finds how many terms it takes 
to improve on drop me a line and tell me that, too. 

Of course, all this may disappoint you. To be sure, the 
endless series is a mathematical representation of the true 
and exact value of tt- To a mathematician, it is as valid a 
way as any to express that value. But if you want it in the 
form of an actual number, how does it help you? It isn’t 
even practical to sum up a couple of dozen terms for any- 
one who wants to go about the ordinary business of living; 
how, then, can it be possible to sum up an infinite number? 

Ah, but mathematicians do not give up on the sum of 
a series just because the number of terms in it is unending. 
For instance, the series: 

¥2 + 14 + % + + V?. ( > + VvA ■ ■ . 

can be summed up, using successively more and more 
terms. If you do this, you will find that the more terms 
you use, the closer you get to 1 , and you can express this 
in shorthand form by saying that the sum of that infinite 
number of terms is merely 1 after all. 

There is a formula, in fact, that can be used to deter- 
mine the sum of any decreasing geometric progression, of 
which the above is an example. 



96 


NUMBERS AND MATHEMATICS 


A Piece of Pi 


97 


Thus, the series: 

/4o + + 9'5 000 H-%0000 + % 00000 - . . 

adds up, in all its splendidly infinite numbers, to a mere, 
and the series: 

/is + Wo + 1 o + / i o o (i bj oo f 1 1 > . . . 

adds up to %. 

To be sure, the series worked out for the evaluation of 
7 T are none of them decreasing geometric progressions, 
and so the formula cannot be used to evaluate the sum. In 
fact, no formula has ever been found to evaluate the sum 
of the Leibniz series or any of the others. Nevertheless, 
there seemed no reason at first to suppose that there might 
not be some way of finding a decreasing geometric pro- 
gression that would evaluate 77. If so, 77 would then be ex- 
pressible as a fraction. A fraction is actually the ratio of 
two numbers and anything expressible as a fraction, or 
ratio, is a ‘ rational number/ * as I explained in the previ- 
ous chapter. The hope /then, was that tt might be a rational 
number. 

One way of proving that a quantity is a rational number 
is to work out its value decimally as far as you can (by 
adding up more and more terms of an infinite series, for 
instance) and then show the result to be a “repeating 
decimal ; that is, a decimal in which digits or some group 
of digits repeat themselves endlessly. 

For instance, the decimal value of H is 0.33333333333 
« • • > while that of !-t is 0.142857 142857 142857 
and so on endlessly. Even a fraction such as Vs which 
seems to “come out even” is really a repeating decimal if 
you count zeros, since its decimal equivalent is 0.125000- 
000000. ... It can be proved mathematically that every 
fraction, however complicated, can be expressed as a deci- 
mal which sooner or later becomes a repeating one. Con- 
versely, any decimal which ends by becoming a repeating 
one, however involved the repetitive cycle, can be ex- 
pressed as an exact fraction. 

Take any repeating decimal at random, say 0.37373737- 


373737. . . . First, you can make a decreasing geometrical 
progression out of it by writing it as: 

3 7 ^oo + 3 ? 4 nooi*-f ^rioooftOO + ^ooooniHm ■ . . 

and you can then use the formula to work out its sum, 
which comes out to (Work out the decimal equivalent 
of that fraction and see what you get.) 

Or suppose you have a decimal which starts out non- 
repetitively and then becomes repetitive, such as 15.216- 
55555555555. . . . This can be written as: 

1 5 + 2 1( ri 000 4 - T .h 0000 + ) 00000 + tioooooo . . . 

From ^ioooo on, wc have a decreasing geometric pro- 
gression and its sum works out to be %ooo. So the series 
becomes a finite one made out of exactly three terms and 
no more, and can be summed easily: 

1 5 -f 21 9 iooo -h %oooo “ 1M04 %ooo 

If you wish, work out the decimal equivalent of 13(m %ooo 
and see what you get. 

Well, then, if the decimal equivalent of 77 w T ere worked 
out for a number of decimal places and some repetition 
were discovered in it, however slight and however com- 
plicated, provided it could be shown to go on endlessly, a 
new series could be written to express its exact value. This 
new series would conclude with a decreasing geometric 
progression which could be summed. There would then be 
a finite series and the true value of x could be expressed 
not as a series but as an actual number. 

Mathematicians threw themselves into the pursuit. In 
1593 Vieta himself used his own series to calculate x to 
seventeen decimal places. Here it is, if you want to stare 
at it: 3.14159265358979323. As you see, there are no 
apparent repetitions of any kind. 

Then in 1615 the German mathematician Ludolf von 
Ceulen used an infinite series to calculate x to thirty-five 
places. He found no signs of repetitiveness, either. How- 
ever, this was so impressive a feat for his time that he 
won a kind of fame, for x is sometimes called “Ludolfs 
number” in consequence, at least in German textbooks. 

And then in 1717 the English mathematician Abraham 



98 


NUMBERS AND MATHEMATICS 


A Piece of Pi 


99 


Sharp went Ludolfs several better by finding x to seventy- 
two decimal places. Still no sign of repeating. 

But shortly thereafter, the game was spoiled. 

To prove a quantity is rational, you have to present the 
fraction to which it is equivalent and display it. To prove 
it is irrational, however, you need not necessarily work 
out a single decimal place. What you must do is to suppose 
that the quantity can be expressed by a fraction, pf q } and 
then demonstrate that this involves a contradiction, such 
as that p must at the same time be even and odd. This 
would prove that no fraction could express the quantity, 
which would therefore be irrational. 

Exactly this sort of proof was developed by the ancient 
Greeks to show that the square root of 2 was an irrational 
number (the first irrational ever discovered). The Pytha- 
goreans were supposed to have been the first to discover 
this and to have been so appalled at finding that there 
could be quantities that could not be expressed by any 
fraction, however complicated, that they swore themselves 
to secrecy and provided a death penalty for snitching. But 
like all scientific secrets, from irrationals to atom bombs, 
the information leaked out anyway. 

Well, in 1761 German physicist and mathematician 
Johann Heinrich Lambert finally proved that x was irra- 
tional. Therefore, no pattern at all was to be expected, no 
matter how slight and no matter how many decimal places 
were worked out. The true value can only be expressed as 
an infinite series. 

Alas! 

But shed no tears. Once x was proved irrational, mathe- 
maticians were satisfied. The problem was over. And as 
for the application of 7 r to physical calculations, that prob- 
lem was over and done with, too. You may think that 
sometimes in very delicate calculations it might be neces- 
sary to know 7 T to a few dozen or even to a few hundred 
places, but not so! The delicacy of scientific measurements 
is wonderful these days, but still there are few that ap- 
proach, say, one part in a billion, and for anything that 
accurate which involves the use of 77 , nine or ten decimal 
places would be ample. 

For example, suppose you drew a circle ten billion 
miles across, with the sun at the center, for the purpose 
of enclosing the entire solar system, and suppose you 


wanted to calculate the length of the circumference of this 
circle (which would come to over thirty-one billion miles) 
by using 35I His as the approximate value of x. You would 
be off by less than three thousand miles. 

But suppose you were so precise an individual that you 
found an error of three thousand miles in 31,000,000,000 
to be insupportable. You might then use Ludolfs value of 
x to thirty-five places. You would then be off by a distance 
that would be equivalent to a millionth of the diameter of 
a proton. 

Or let’s take a big circle, say the circumference of the 
known universe. Suppose large radio telescopes under con- 
struction can receive signals from a distance as great as 
40,000,000,000 light-years. A circle about a universe with 
such a radius would have a length of, roughly, 150,000,- 
000,000,000,000,000,000 (150 sextillion) miles. If the 
length of this circumference were calculated by Ludolfs 
value of x to thirty-five places, it would be off by less than 
a millionth of an inch. 

What can one say then about Sharp’s value of x to 
seventy-two places? 

Obviously, the value of x, as known by the time its irra- 
tionality was proven, was already far beyond the accuracy 
that could conceivably be demanded by science, now or 
in the future. 

And yet with the value of x no longer needed for scien- 
tists, past what had already been determined, people never- 
theless continued their calculations through the first half 
of the nineteenth century. 

A fellow called George Vega got x to 140 places, an- 
other called Zacharias Dase did it to 200 places, and some- 
one called Recher did it to 500 places. 

Finally, in 1873 William Shanks reported the value of 
x to 707 places, and that, until 1949, was the record — and 
small wonder. It took Shanks fifteen years to make the 
calculation and, for what that’s worth, no signs of any 
repetitiveness showed up. 

We can wonder about the motivation that would cause 
a man to spend fifteen years on a task that can serve no 
purpose. Perhaps it is the same mental attitude that will 
make a man sit on a flagpole or swallow goldfish in order 
to “break a record,” Or perhaps Shanks saw this as his one 
road to fame. 

If so, he made it. Histories of mathematics, in among 



100 


NUMBERS AND MATHEMATICS 


their descriptions of the work of men like Archimedes, 
Fermat, Newton, Euler, and Gauss, will also find r^om 
for a line to the effect that William Shanks in the years 
preceding 1873 calculated 7 r to 707 decimal places. So 
perhaps he felt that his life had not been wasted. 

But alas, for human vanity — 

In 1949 the giant computers were coming into their 
own, and occasionally the young fellows at the controls, 
full of fun and life and beer, could find time to play with 
them. 

So, on one occasion, they pumped one of the unending 
series into the machine called ENIAC and had it calculate 
the value of x. They kept it at the task for seventy hours, 
and at the end of that time they had the value of tt (shades 
of Shanks!) to 2035 places. * 

And to top it all off for poor Shanks and his fifteen 
wasted years, an error was found in the five hundred 
umpty-umpth digit of Shanks’ value, so that all the digits 
after that, well over a hundred, were wrong! 

And of course, in case you’re wondering, and you 
shouldn’t, the values as determined by computers showed 
no signs of any repetitiveness either. 


* By 1955 a faster computer calculated tt to 10,017 piaees in thirty-three 
hours and , actually , there are interesting mathematical points to be de- 
rived from studying the various digits of tt [ and it’s possible that more 
has been done since, but I haven't kept track]. 


7 TOOLS 
OF THE 
TRADE 


The previous chapter does not conclude the story of 
tt. As the title stated, it was only a piece of tt- Let us 
therefore continue onward. 

The Greek contribution to geometry consisted of idealiz- 
ing and abstracting it. The Egyptians and Babylonians 
solved specific problems by specific methods but never 
tried to establish general rules. 

The Greeks, however, strove for the general and felt 
that mathematical figures had certain innate properties 
that were eternal and immutable. They felt also that a 
consideration of the nature and relationships of these 
properties was the closest man could come to experiencing 
the sheer essence of beauty and divinity. (If I may veer 
away from science for a moment and invade the sacred 
precincts of the humanities, I might point out that just this 
notion was expressed by Edna St. Vincent Millay in a 
famous line that goes : “Euclid alone has looked on Beauty 
bare.”) 

Well, in order to get down to the ultimate bareness of 
Beauty, one had to conceive of perfect, idealized figures 
made up of perfect idealized parts. For instance, the ideal 
line consisted of length and nothing else. It had neither 


101 



102 


NUMBERS AND MATHEMATICS 


Tools of the Trade 


103 


EUCLID 

After the death of Alexander the Great , various of his 
generals seized control of the ancient world , One of them , 
Ptolemy , established a dynasty that was to rule over 
Egypt for three centuries . He converted his capital at 
Alexandria into the greatest intellectual center of ancient 
times , and one of the first luminaries to work there was 
the mathematician Euclid . 

Very little is known about Euclid’s personal life . He 
was born about 325 B.c., we don't know where , and the 
time and place of his death are unknown . 

His name is indissolubly linked to geometry, for he 
wrote a textbook (Elements) on the subject that has been 
standard, with some modifications, of course , ever since . 
// went through more than a thousand editions after the 
invention of printing , and he is undoubtedly the most 
successful textbook writer of all time , 

And yet, as a mathematician, Euclid’s fame is not due 
to his own research. Few of the theorems in his textbook 
are his own . What Euclid did , and what made him great, 
was to take all the knowledge accumulated in mathematics 
to his time and codify it into a single work . In doing so, 
he evolved, as a starting point , a series of axioms and 
postulates that were admirable for their brevity and 
elegance . 

In addition to geometry, his text took up ratio and 
proportion and what is now known as the theory of 
numbers , He made optics a part of geometry , too, by 
dealing with light rays as though they were straight lines . 

One story told about him involves King Ptolemy who 
war studying geometry and who asked if Euclid couldn't 
make his demonstrations a little easier to follow. Euclid 
said, uncompromisingly, " There is no royal road to 
geometry 



The Granger Collection 



104 


NUMBERS AND MATHEMATICS 


Tools of the Trade 


105 


thickness nor breadth nor anything, in fact, but length. 
Two ideal lines, ideally and perfectly straight, intersected 
at an ideal and perfect point, which had no dimensions at 
all, only position. A circle was a line that curved in per- 
fectly equal fashion at all points; and every point on that 
curve was precisely equally distant from a particular point 
called the center of the circle. 

Unfortunately, although one can imagine such abstrac- 
tions, one doesn’t usually communicate them as abstrac- 
tions alone. In order to explain the properties of such 
figures (and even in order to investigate them on your 
own) it is helpful, almost essential in fact, to draw crass, 
crude, and ungainly approximations in wax, on mud, on 
blackboard, or on paper, using a pointed stick, chalk, 
pencil, or pen. (Beauty must be swathed in drapery in 
mathematics, alas, as in life.) 

Furthermore, in order to prove some of the ineffably 
beautiful properties of various geometrical figures, it was 
usually necessary to make use of more lines than existed 
in the figure alone. It might be necessary to draw a new 
line through a point and make it parallel or, perhaps, 
perpendicular to a second line. It might be necessary to 
divide a line into equal parts, or to double the size of an 
angle. 

To make all this drawing as neat and as accurate as 
possible, instruments must be used. It follows naturally, I 
think, once you get into the Greek way of thinking, that 
the fewer and simpler the instruments used for the pur- 
pose, the closer the approach to the ideal. 

Eventually, the tools were reduced to an elegant mini- 
mum of two. One is a straightedge for the drawing of 
straight lines. This is not a ruler, mind you, with inches 
or centimeters marked off on it. It is an unmarked piece 
of wood (or metal or plastic, for that matter) which can 
do no more than guide the marking instrument into the 
form of a straight line. 

The second tool is the compass, which, while most sim- 
ply used to draw circles, will also serve to mark off equal 
segments of lines, will draw intersecting arcs that mark a 
point that is equidistant from two other points, and so on. 

I presume most of you have taken plane geometry and 
have utilized these tools to construct one line perpendicu- 
lar to another, to bisect an angle, to circumscribe a circle 
about a triangle, and so on. All these tasks and an infinite 


number of others can be performed by using the straight- 
edge and compass in a finite series of manipulations. 

By Plato’s time, of course, it was known that by using 
more complex tools, certain constructions could be sim- 
plified; and, in fact, that some constructions could be per- 
formed which, until then, could not be performed by 
straightedge and compass alone. That, to the Greek geome- 
ters, was something like shooting a fox or a sitting duck, 
or catching fish with worms, or looking at the answers in 
the back of the book. It got results but it just wasn’t the 
gentlemanly thing to do. The straightedge and compass 
were the only “proper” tools of the geometrical trade. 

Nor was it felt that this restriction to the compass and 
straightedge unduly limited the geometer. It might be tedi- 
ous at times to stick to the tools of the trade; it might be 
easier to take a short cut by using other devices; but surely 
the straightedge and compass atone could do it all, if you 
were only persistent enough and ingenious enough. 

For instance, if you are given a line of a fixed length 
which is allowed to represent the numeral 1 , it is possible 
to construct another line, by compass and straightedge 
alone, exactly twice that length to represent 2, or another 
line to represent 3 or 5 or 500 or Vi or Vi or or % or 2% 
or 27 1( &. In fact, by using compass and straightedge only, 
any rational number (i.e., any integer or fraction) could 
be duplicated geometrically. You could even make use of 
a simple convention (which the Greeks never did, alas) 
to make it possible to represent both positive and negative 
rational numbers. 

Once irrational numbers were discovered, numbers for 
which no definite fraction could be written, it might seem 
that compass and straightedge would fail, but even then 
they did not. 

For instance, the square root of 2 has the value 
1.414214 . . . and on and on without end. How, then, can 
you construct one line which is 1.414214 . . , times as 
long as another when you cannot possibly ever know ex- 
actly how many times as long you want it to be. 

Actually, it’s easy. Imagine a given line from point A 
to point B . (I can do this without a diagram, I think, but 
if you feel the need you can sketch the lines as you read. 
It won’t be hard.) Let this line, AB t represent L . 

Next, construct a line at B , perpendicular to AB . Now 
you have two lines forming a right angle. Use the compass 



106 NUMBERS AND MATHEMATICS 

to draw a circle with its center at B, where the two lines 
meet, and passing through A. It will cut the perpendicular 
line you have just drawn at a point we can call C. Because 
of the well-known properties of the circle, line BC is ex- 
actly equal to line AB, and is also 1 . 

Finally, connect points A and C with a third straight 
line. 

That line, AC, as can be proven by geometry, is exactly 

V 2 times as long as either AB or BC , and therefore rep- 
resents the irrational quantity y" 2. 

Don’t, of course, think, that it is now only necessary to 
measure AC in terms of AB to obtain an exact value of 

V 2. The construction was drawn by imperfect instru- 

ments in the hands of imperfect men and is only a crude 
approximation of the ideal figures they represent. It is the 
ideal line represented by AC that is and not AC 

itself in actual reality. 

It is possible, in similar fashion, to use the straightedge 
and compass to represent an infinite number of other irra- 
tional quantities. 

In fact, the Greeks had no reason to doubt that any 
conceivable number at all could be represented by a line 
that could be constructed by use of straightedge and com- 
pass alone in a finite number of steps. And since all con- 
structions boiled down to the construction of certain lines 
representing certain numbers, it was felt that anything that 
could be done with any tool could be done by straightedge 
and compass alone. Sometimes the details of the straight- 
edge and compass construction might be elusive and re- 
main undiscovered, but eventually, the Greeks felt, given 
enough ingenuity, insight, intelligence, intuition, and luck, 
the construction could be worked out. 

For instance, the Greeks never learned how to divide a 
circle into seventeen equal parts by straightedge and com- 
pass alone. Yet it could be done. The method was not 
discovered until 1801, but in that year, the German mathe- 
matician Karl Friedrich Gauss, then only twenty-four, 
managed it. Once he divided the circle into seventeen 
parts, he could connect the points of division by a straight- 
edge to form a regular polygon of seventeen sides (a 
“septendecagon”). The same system could be used to 
construct a regular polygon of 257 sides, and an infinite 
number of other polygons with still more sides, the num- 


Tools of the Trade 107 

ber of sides possible being calculated by a formula which 
I won’t give here. 

If the construction of a simple thing like a regular sep- 
tendecagon could elude the great Greek geometers and 
yet be a perfectly soluble problem in the end, why could 
not any conceivable construction, however puzzling it 
might seem, yet prove soluble in the end. 

As an example, one construction that fascinated the 
Greeks was this: Given a circle, construct a square of the 
same area. 

This is called “squaring the circle.” 

There are several ways of doing this. Here’s one method. 
Measure the radius of the circle with the most accurate 
measuring device you have — say, just for fun, that the 
radius proves to be one inch long precisely. (This method 
will work for a radius of any length, so why not luxuriate 
in simplicity.) Square that radius, leaving the value still 1, 
since lxl is 1 , thank goodness, and multiply that by the 
best value of n you can find. (Were you wondering when 
I’d get back to x?) If you use 3.1415926 as your value 
of 7 t , the area of the circle proves to be 3.1415926 square 
inches. 

Now, take the square root of that, which is 1.7724539 
inches, and draw a straight line exactly 1.7724539 inches 
long, using your measuring device to make sure of the 
length. Construct a perpendicular at each end of the line, 
mark off 1.7724539 inches on each perpendicular, and 
connect those two points. 

Voila! You have a square equal in area to the given 
circle. Of course, you may feel uneasy. Your measuring 
device isn’t infinitely accurate and neither is the value of ^ 
which you used. Does not this mean that the squaring of 
the circle is only approximate and not exact? 

Yes, but it is not the details that count but the principle. 
We can assume the measuring device to be perfect, and 
the value of tt which was used to be accurate to an infinite 
number of places. After all, this is just as justifiable as 
assuming our actual drawn lines to represent ideal lines, 
considering our straightedge perfectly straight and our 
compass to end in two perfect points. In principle, we have 
indeed perfectly squared the circle. 

Ah, but we have made use of a measuring device, which 
is not one of the only two tools of the trade allowed a 



108 


NUMBERS AND MATHEMATICS 


KARL FRIEDRICH GAUSS 

Gauss, the son of a gardener , was bom in Braunschweig f 
Germany , on April 30, 1777 . He an infant prodigy 
in mathematics who remained a prodigy all his life . He 
was capable of great feats of memory and of mental 
calculation . At the age of three , he was already correcting 
his father’s sums. His unusual mind was recognized and 
he was educated at the expense of Duke Ferdinand of 
Brunswick. In 1795 Gauss entered the University of Got- 
tingen . 

While still in his teens he made a number of remark- 
able discoveries f including the (t method of least squares,” 
which could determine the best curve fitting a group of 
as few as three observations . While still in the univer- 
sity, he demonstrated a method for constructing an equi- 
lateral polygon of seventeen sides and , more important, 
showed which polygons would not be so constructed — 
the first demonstration of a mathematical impossibility. 

In 1799 Gauss proved the fundamental theorem of 
algebra, that every algebraic equation has a root in the 
form of a complex number , and in 1801 he went on to 
prove the fundamental theorem of arithmetic, that every 
natural number can be represented as the product of 
primes in one and only one way . 

All this required intense concentration. There is a 
story that when he was told in 1807 that his wife was 
dying, he looked up from the problem that was engaging 
him and muttered, “Tell her to wait a moment till Fm 
through /' 

His agile mind never seemed to cease . At the age of 
sixty -two he taught himself Russian, Personal tragedy 
dogged him , though , Each of his two wives died young, 
and only one of his six children survived him. He died 
in Gottingen on February 23, 1855 . 



The Granger Collection 



110 NUMBERS AND MATHEMATICS 

gentleman geometer. That marks you as a cad and bounder 
and you are hereby voted out of the club. 

Here’s another method of squaring the circle. What you 
really need, assuming the radius of your circle to represent 
1, is another straight line representing A square built 
on such a line would have just the area of a unit-radius 
circle. How to get such a line? Well, if you could con- 
struct a line equal to tt times the length of the radius, 
there are known methods, using straightedge and compass 
alone, to construct a line equal in length to the square 
root of that line, hence representing the which we 
are after. 

But it is simple to get a line that is w times the radius. 
According to a well-known formula, the circumference of 
the circle is equal in length to twice the radius times w. 
So let us imagine the circle resting on a straight line and 
let’s make a little mark at the point where the circle just 
touches the line. Now slowly turn the circle so that it 
moves along the line (without slipping) until the point 
you have marked makes a complete circuit and once again 
touches the line. Make another mark where it again 
touches. Thus, you have marked off the circumference of 
the circle on a straight line and the distance between the 
two marks is twice ?r. 

Bisect that marked-off line by the usual methods of 
straightedge and compass geometry and you have a line 
representing w . Construct the square root of that line and 
you have y^7 

Voila! By that act, you have, in effect, squared the 
circle. 

But no. I’m afraid you’re still out of the club. You have 
made use of a rolling circle with a mark on it and that 
comes under the heading of an instrument other than the 
straightedge and compass. 

The point is that there are any number of ways of 
squaring the circle, but the Greeks were unable to find any 
way of doing it with straightedge and compass alone in a 
finite number of steps. (They spent I don’t know how 
many man-hours of time searching for a method, and look- 
ing back on it, it might all seem an exercise in futility 
now, but it wasn’t. In their search, they came across all 
sorts of new curves, such as the conic sections, and new 
theorems, which were far more valuable than the squar- 
ing of the circle would have been.) 


Tools of the T rade 111 


Although the Greeks failed to find a method, the search 
continued and continued. People kept on trying and trying 
and trying and trying — 

And now let’s change the subject for a while. 


Consider a simple equation such as 2x— 1=0. You can 
see that setting x—% will make a true statement out of it, 
for 206) — 1 is indeed equal to zero. No other number 
can be substituted for * in this equation and yield a true 
statement. 

By changing the integers in the equation (the “coeffi- 
cients” as they are called) * can be made to equal other 
specific numbers. For instance, in 3x— 4=0, x is equal to 
Ys; and in 7 jc+2=0, x=—% In fact, by choosing the 
coefficients appropriately, you can have as a value of x 
any positive or negative integer or fraction whatever. 

But in such an “equation of the first degree,” you can 
only obtain rational values for You can’t possibly have 
an equation of the form (Ax-j-B— 0), where A and B are 
rational, such that x will turn out to be equal to V 2, for 
instance. 

The thing to do is to try a more complicated variety of 
equation. Suppose you try x 2 — 2=0, which is an “equation 
of the second degree” because it involves a square. If you 
solve for x you’ll find the answer, Y 2, when substituted 
for jc will yield a true statement. In fact, there are two 
possible answers, for the substitution of — V 2 for x will 
also yield a true statement. 

You can build up equations of the third degree, such as 
Ax fi +Bx 2 +Cx+D=Q, or of the fourth degree (I don’t 
have to give any more examples, do I?), or higher. Solving 
for x in each case becomes more and more difficult, but 
will give solutions involving cube roots, fourth roots, and 
so on. 

In any equation of this type (a “polynomial equation”) 
the value of jc can be worked out by manipulating the 
coefficients. To take the simplest case, in the general equa- 
tion of the first degree: Ax+B— 0, the value of x is —B!A. 
In the general equation of the second degree : 

Ax 2 -\-Bx+C= 0, there are two solutions. One is 


-B+y B 2 —4AC 
2 A 


and the other is 


~B-^B 2 -4AC 


2 A 



112 


NUMBERS AND MATHEMATICS 


Solutions get progressively more complicated and even- 
tually, for equations of the fifth degree and higher, no 
general solution can be given, although specific solutions 
can still be worked out. The principle remains, however, 
that in all polynomial equations, the value of x can be 
expressed by use of a finite number of integers involved 
in a finite number of operations, these operations consist- 
ing of addition, subtraction, multiplication, division, rais- 
ing to a power f ‘Involution"), and extracting roots (“evo- 
lution”). 

These operations are the only ones used in ordinary 
algebra and are therefore called “algebraic operations." 
Any number which can be derived from the integers by a 
finite number of algebraic operations in any combination 
is called an “algebraic number,'’ To put it in reverse, any 
algebraic number is a possible solution for some poly- 
nomial equation. 

Now it so happens that the geometric equivalent of all 
the algebraic operations, except the extraction of roots 
higher than the square root, can be performed by straight- 
edge and compass alone. If a given line represents 1, there- 
fore, it follows that a line representing any algebraic num- 
ber that involves no root higher than the square root can 
be constructed by straightedge and compass in a finite 
number of manipulations. 

Since x does not seem to contain any cube roots (or 
worse) , is it possible that it can be constructed by straight- 
edge and compass? That might be if algebraic numbers 
included all numbers. But do they? Are there numbers 
which cannot be solutions to any polynomial equation, 
and are therefore not algebraic? 

To begin with, all possible rational numbers can be 
solutions to equations of the first degree, so all rational 
numbers are algebraic numbers. Then, certainly some irra- 
tional numbers are algebraic numbers, for it is easy to 
write equations for which -\J 2 or ^15 — 3 are solutions. 

But can there be irrational numbers which will not serve 
as a solution to a single one of the infinite number of 
different polynomial equations in each of all the infinite 
number of degrees possible? 

In 1844 the French mathematician Joseph Liouville 
finally found a way of showing that such nonalgebraic 
numbers did exist. (No, I don’t know how he did it, but 
if any reader thinks I can understand the method, and I 


Tools of the Trade 113 

must warn him not to overestimate me, he is welcome to 
send it in.) 

However, having proved that nonalgebraic numbers 
existed, LiouviKe could still not find a specific example. 
The nearest he came was to show that a number repre- 
sented by the symbol e could not serve as the root for 
any conceivable equation of the second degree. 

(At this point I am tempted to launch into a discussion 
of the number e because, as I said at the start of the previ- 
ous chapter, there is the famous equation — — 1. But I’ll 
resist temptation because, for one thing, I had some things 
to say about e in Chapter 3.) 

Then, in 1873, the French mathematician Charles Her- 
mite worked out a method of analysis that showed that e 
could not be the root of any conceivable equation of any 
conceivable degree and hence was actually not an alge- 
braic number. It was, in fact, what is called a “transcen- 
dental number,” one which transcends (that is, goes 
beyond) the algebraic operations and cannot be produced 
from the integers by any finite number of those operations. 
(That is, \[2 is irrational but can be produced by a single 
algebraic operation, taking the square root of 2. The value 
of e, on the other hand, can only be calculated by the use 
of infinite series involving an infinite number of additions, 
divisions, subtractions, and so on.) 

Using the methods developed by Hermite, the German 
mathematician Ferdinand Lindemann in 1882 proved that 
x, too, was a transcendental number. 

This is crucial for the purposes of this chapter, for it 
meant that a line segment equivalent to x cannot be built 
up by the use of the straightedge and compass alone in a 
finite number of manipulations. The circle cannot be 
squared by straightedge and compass alone . It is as im- 
possible to do this as to find an exact value for \/^ or 
to find an odd number that is an exact multiple of 4. 

One odd point about transcendental numbers 

They were difficult to find, but now that they have been, 
they prove to be present in overwhelming numbers. Prac- 
tically any expression that involves either e or ir is tran- 
scendental, provided the expression is not arranged so that 
the e or x cancel out. Practically all expressions involving 
logarithms (which involve e) and practically all expres- 
sions involving trigonometric functions (which involve x) 



114 


NUMBERS AND MATHEMATICS 


are transcendental. Expressions involving numbers raised 
to an irrational power, such as x v3 , are transcendental. 

In fact, if you refer back to Chapter 5, you will under- 
stand me when I say that it has been proved that the alge- 
braic numbers can be put into one-to-one correspondence 
with the integers, but the transcendental numbers cannot. 
This means that the algebraic numbers, although infinite, 
belong to the lowest of the transfinite numbers, ^ 0 , while 
the transcendental numbers belong, at the least, to the next 
higher transfinite, There are thus infinitely more tran- 
scendental numbers than there are algebraic numbers. 

To be sure, the fact that the transcendentality of tt is 
now well established and has been for nearly a century 
doesn’t stop the ardent circle-squarers, who continue to 
work away desperately with straightedge and compass and 
continue to report solutions regularly. 

So if you know a way to square the circle by straight- 
edge and compass alone, I congratulate you, but you have 
a fallacy in your proof somewhere. And it’s no use send- 
ing it to me, because I’m a rotten mathematician and 
couldn’t possibly find the fallacy, but I tell you anyway, 
it’s there. 


8 THE 

IMAGINARY 
THAT ISN'T 


When I was a mere slip of a lad and attended col- 
lege, I had a friend with whom I ate lunch every day. 
His 11 a.m. class was in sociology, which I absolutely 
refused to take, and my 11 a.m. class was calculus, which 
he as steadfastly refused to take — so we had to separate 
at eleven and meet at twelve. 

At it happened, his sociology professor was a scholar 
who did things in the grand manner, holding court after 
class was over. The more eager students gathered close 
and listened to him pontificate for an additional fifteen 
minutes, while they threw in an occasional log in the 
form of a question to feed the flame of oracle. 

Consequently, when my calculus lecture was over, I 
had to enter the sociology room and wait patiently for 
court to conclude. 

Once I walked in when the professor was listing on the 
board his classification of mankind into the two groups 
of mystics and realists, and under mystics he had included 
the mathematicians along with the poets and theologians. 
One student wanted to know why. 

“Mathematicians,’* said the professor, “are mystics be- 
cause they believed in numbers that have no reality.” 

Now ordinarily, as a nonmember of the class, I sat in 
the corner and suffered in silent boredom, but now I rose 
convulsively, and said, “What numbers?” 


115 



116 


NUMBERS AND MATHEMATICS 


The professor looked in my direction and said, “The 
square root of minus one. It has no existence. Mathema- 
ticians call it imaginary. But they believe it has some kind 
of existence in a mystical way.” 

‘There’s nothing mystical about it/ 1 I said, angrily. 
l The square root of minus one is just as real as any other 
number.” 

The professor smiled, feeling he had a live one on 
whom he could now proceed to display his superiority of 
intellect (I have since had classes of my own and I know 
exactly how he felt). He said, silkily, “We have a young 
mathematician here who wants to prove the reality of the 
square root of minus one. Come, young man, hand me 
the square root of minus one pieces of chalk!” 

I reddened, “Well, now, wait — ” 

“That’s all,” he said, waving his hand. Mission, he 
imagined, accomplished, both neatly and sweetly. 

But I raised my voice. ‘Til do it. I’ll do it. I’ll hand 
you the square root of minus one pieces of chalk, if you 
hand me a one-half piece of chalk.” 

The professor smiled again, and said, “Very well,” 
broke a fresh piece of chalk in half, and handed me one 
of the halves, “Now for your end of the bargain.” 

“Ah, but wait,” I said, “you haven’t fulfilled your end. 
This is one piece of chalk you’re handed me, not a one- 
half piece.” I held it up for the others to see. “Wouldn’t 
you all say this was one piece of chalk? It certainly isn’t 
two or three,” 

Now the professor wasn’t smiling. “Hold it. One piece 
of chalk is a piece of regulation length. You have one 
that’s half the regulation length.” 

I said, “Now you’re springing an arbitrary definition 
on me. But even if I accept it, are you willing to main- 
tain that this is a one-half piece of chalk and not a 0.48 
piece or a 0.52 piece? And can you really consider yourself 
qualified to discuss the square root of minus one, when 
you’re a little hazy on the meaning of one half?” 

But by now the professor had lost his equanimity al- 
together and his final argument was unanswerable. He 
said, “Get the hell out of here!” I left (laughing) and 
thereafter waited for my friend in the corridor. 

Twenty years have passed since then and I suppose I 
ought to finish the argument — 


The Imaginary That Isn't 


117 


Let’s start with a simple algebraic equation such as 
^3— 5. The expression, x, represents some number 
which, when substituted for x, makes the expression a 
true equality. In this particular case x must equal 2, 
since 2+3 — 5, and so we have “solved for x” 

The interesting thing about this solution is that it is 
the only solution. There is no number but 2 which will 
give 5 when 3 is added to it. 

This is true of any question of this sort, which is called 
a “linear equation” (because in geometry it can be repre- 
sented as a straight line) or “a polynomial equation of 
the first degree.” No polynomial equation of the first 
degree can ever have more than one solution for x. 

There are other equations, however, which can have 
more than one solution. Here’s an example: x 2 — 5x+6^0, 
where x 2 (“x square” or “x squared”) represents x times 
x. This is called a “quadratic equation,” from a Latin 
word for “square,” because it involves x square. It is also 
called “a polynomial equation of the second degree” be- 
cause of the little 2 in x 2 . As for x itself, that could be 
written x 1 except that the 1 is always omitted and taken 
for granted, and that is why x+3— 5 is an equation of 
the first degree. 

If we take the equation x 2 — 5x+6— 0, and substitute 
2 for x, then x 2 is 4, while 5x is 10, so that the equation 
becomes 4—10+6—10, which is correct, making 2 a 
solution of the equation. 

However, if we substitute 3 for x, then x 2 is 9 and 5x 
is 15, so that the equation becomes 9—15+6—0, which is 
also correct, making 3 a second solution of the equation. 

Now no equation of the second degree has ever been 
found which has more than two solutions, but what about 
polynomial equations of the third degree? These are 
equations containing x 3 (“x cube” or “x cubed”), which 
are therefore also called “cubic equations.” The expression 
x 3 represents x times x times x. 

The equation x 3 — 6x 2 +llx— 6— 0 has three solutions, 
since you can substitute 1, 2, or 3 for x in this equation 
and come up with a true equality in each case. No cubic 
equation has ever been found with more than three solu- 
tions, however. 

In the same way polynomial equations of the fourth 
degree can be constructed which have four solutions but 
no more; polynomial equations of the fifth degree, which 



118 


NUMBERS AND MATHEMATICS 


have five solutions but no more; and so on. You might 
say, then, that a polynomial equation of the nth degree 
can have as many as n solutions, but never more than n . 

Mathematicians craved something even prettier than 
that and by about 1800 found it. At that time, the Ger- 
man mathematician Karl Friedrich Gauss showed that 
every equation of the nth degree had exactly n solutions, 
not only no more, but also no less. 

However, in order to make the fundamental theorem 
true, our notion of what constitutes a solution to an alge- 
braic equation must be drastically enlarged. 

To begin with, men accept the “natural numbers” only: 
1, 2, 3, and so on. This is adequate for counting objects 
that are only considered as units generally. You can 
have 2 children, 5 cows, or 8 pots; while to have 2 1 ,& 
children, 5 1 4 cows, or 8*4 pots does not make much sense. 

In measuring continuous quantities such as lengths or 
weights, however, fractions became essential. The Egyp- 
tians and Babylonians managed to work out methods of 
handling fractions, though these were not very efficient by 
our own standards; and no doubt conservative scholars 
among them sneered at the mystical mathematicians who 
believed in a number like 5 % which was neither 5 nor 6. 

Such fractions are really ratios of whole numbers. To 
say a plank of wood is 2% yards long, for instance, is to 
say that the length of the plank is to the length of a 
standard yardstick as 21 is to 8. The Greeks, however, 
discovered that there were definite quantities which could 
not be expressed as ratios of whole numbers. The first to 
be discovered was the square root of 2, commonly ex- 
pressed as \/2, which is that number which, when multi- 
plied by itself, gives 2. There is such a number but it 
cannot be expressed as a ratio; hence, it is an “irrational 
number.” 

Only thus far did the notion of number extend before 
modern times. Thus, the Greeks accepted no number 
smaller than zero. How can there be less than nothing? 
To them, consequently, the equation *+5 = 3 had no 
solution. How can you add 5 to any number and have 3 
as a result? Even if you added 5 to the smallest number 
(that is, to zero), you would have 5 as the sum, and if 
you added 5 to any other number (which would have to 


The Imaginary That Isn't 119 

be larger than zero), you would have a sum greater 
than 5, 

The first mathematician to break this taboo and make 
systematic use of numbers less than zero was the Italian, 
Girolamo Cardano. After all, there can be less than noth- 
ing. A debt is less than nothing. 

If all you own in the world is a two-dollar debt, you 
have two dollars less than nothing. If you are then given 
five dollars, you end with three dollars of your own 
(assuming you are an honorable man who pays his debts). 
Consequently, in the equation *+5=3, * can be set equal 
to —2, where the minus sign indicates a number less than 
zero. 

Such numbers are called “negative numbers” from a 
Latin word meaning “to deny,” so that the very name 
carries the traces of the Greek denial of the existence of 
such numbers. Numbers greater than zero are “positive 
numbers” and these can be written +1, +2, +3, and 
so on. 

From a practical standpoint, extending the number 
system by including negative numbers simplifies all sorts 
of computations; as, for instance, those in bookkeeping. 

From a theoretical standpoint, the use of negative num- 
bers means that every equation of the first degree has 
exactly one solution. No more; no less. 

If we pass on to equations of the second degree, we 
find that the Greeks would agree with us that the equa- 
tion x 2 — 5*+ 6=0 has two solutions. 2 and 3. They would 
say, however, that the equation x 2 +4x— 5=0 has only 
one solution, 1. Substitute 1 for * and * 2 is 1, while Ax 
is 4, so that the equation becomes 1 + 4— 5=0. No other 
number will serve as a solution, as long as you restrict 
yourself to positive numbers. 

However, the number —5 is a solution, if we consider 
a few rules that are worked out in connection with the 
multiplication of negative numbers. In order to achieve 
consistent results, mathematicians have decided that the 
multiplication of a negative number by a positive number 
yields a negative product, while the multiplication of a 
negative number by a negative number yields a positive 
product. 

If, in the equation * 2 +4*~5=0, ~5 is substituted for 
x t then x 2 becomes —5 times —5, or +25, while Ax be- 
comes +4 times -5, or -20. The equation becomes 



120 


NUMBERS AND MATHEMATICS 


The Imaginary That Isn't 


121 


25—20— 5=0, which is true. We would say, then, that 
there are two solutions to this equation, + 1 and —5. 

Sometimes, a quadratic equation does indeed seem to 
have but a single root, as, for example, * 2 — 6x+9=0, 
which will be a true equality if and only if the number 
4*3 is substituted for x. However, the mechanics of solu- 
tion of the equation show that there are actually two solu- 
tions, which happen to be identical. Thus, x 2 ~ 6* +9=0 
can be converted to (x— 3) (*— 3)=0 and each (x— 3) 
yields a solution. The two solutions of this equation are, 
therefore, +3 and. +3. 

Allowing for occasional duplicate solutions, are we ready 
to say then that all second degree equations can be shown 
to have eactly two solutions if negative numbers are in- 
cluded in the number system? 

Alas, no! For what about the equation **+1=0. To 
begin with, x 2 must be —1 since substituting —1 for 
x 2 makes the equation -1 + 1—0, which is correct enough. 

But if * 2 is — l,Jhen x must be the famous square root 
of minus one (\/— -1), which occasioned the set-to be- 
tween the sociology professor and myself. The square 
root of minus one is that number which when multiplied 
by itself will give —1. But there is no such number in the 
set of positive and negative quantities, and that is the 
reason the sociology professor scorned it First, +1 times 
+ 1 is +1; secondly, —1 times —1 is +1. 

To allow any solution at all for the equation x 2 +l=0, 
let alone two solutions, it is necessary to get past this road- 
block. If no positive number will do and no negative one 
either, it is absolutely essential to define a completely 
new kind of number; an imaginary number, if you like; 
one with its square equal to — 1 . 

We could, if we wished, give the new kind of number 
a special sign. The plus sign does for positives and the 
minus sign for negatives; so we could use an asterisk for 
the new number and say that *1 (“star one”) times *1 
was equal to —1. 

However, this was not done. Instead, the symbol i 
(for “imaginary”) was introduced by the Swiss mathe- 
matician Leonhard Euler in 1777 and was thereafter gen- 
erally adopted. So we can write i=\/ — 1 or p= — 1. 

Having defined i in this fashion, we can express the 
square root of an^Miegative number. For instance, y — 4 
can be written y4 times y — 1, or 2 /. In general, any 


square root of a negative number, y —n f can be written 
as the square root of the equivalent posit ive number 
times the square root of minus one; that is, \Z-n=yni, 

In this way, we can picture a whole series of imaginary 
numbers exactly analogous to the series of ordinary or 

“real numbers.” For 1, 2, 3, 4 we would have 

i f 2 i, 3 i, 4 i. . , , This would include fractions, for % 
would be matched by Wi 7 by and so on. It 

would_also include irrationals, for y2 would be matched 
by V 2 z and even a number like * (pi) would be matched 
by W, 

These are all comparisons of positive numbers with 
imaginary numbers. What about negative numbers? Well, 
why not negative imaginaries, too? For —1, —2, —3, 
—4, . . , , there would be — /, —2i, —3 i t — , . . 

So now we have four classes of numbers: 1) positive 
real numbers, 2) negative real numbers, 3 ) positive 
imaginary numbers, 4) negative imaginary numbers. 
(When a negative imaginary is multiplied by a negative 
imaginary, the product is negative.) 

Using this further extension of the number system, we 
can find the necessary two solutions for the equation 
*2+1=0. They are +i and — /. First +i times +i equals 

— 1, and secondly —i times — i equals — 1, so that in 
either case, the equation becomes —1 + 1=0, which is a 
true equality. 

In fact, you can use the same extension of the number 
system to find all four solutions for an equation such as 
x 4 —l=0. The solutions are +1, —1, +*, and — i. To 
show this, we must remember that any number raised to 
the fourth power is equal to the square of that number 
multiplied by itself. That is, n 4 equals n 2 times n 2 . 

Now let's substitute each of the suggested solutions 
into the equations so that x 4 becomes successively ( + 1) 4 , 
(-1) 4 , ( + 0 4 , and (-i)K 

First ( + 1) 4 equals ( + 1) 2 times ( + 1) 2 , and since 
( + 1) 2 equals +1, that becomes +1 times +1, which 
is +1. 

Second, ( — l) 4 equals ( — l) 2 times ( — l) 2 , and since 
( — l) 2 also equals +1, the expression is again +1 times 
+ 1 , or + 1 . 

Third, ( + i) 4 equals ( + 0 2 times (+i) 2 and we have 
defined (+i) 2 as —1, so that the expression becomes 

— 1 times —1, or +1. 



122 


numbers and mathematics 


The Imaginary That Isn't 


123 


LEONHARD EULER 


Euler , the son of a Calvinist minister, was born in Basel, 
Switzerland , on April 15, 1707 * He received his masters 
degree at the age of sixteen from the University of Basel . 

Euler went to St . Petersburg, Russia, in 1727, for 
there Catherine 1 (the widow of Peter the Great) had 
recently founded the Petersburg Academy and Euler spent 
much of his life there . In 1735 he lost the sight of his 
right eye through too-ardent observations of the Sun in 
an attempt to work out a system of time determination . 

In 1741 Euler went to Berlin to head and revivify the 
decaying Academy of Sciences but didn't get along with 
the new Prussian King, Frederick 11. He returned to St. 
Petersburg in 1766 and there he died on September 18, 

1783 . fJ . 

Euler was the most prolific mathematician of all time, 
writing on every branch of the subject and being always 
careful to describe his reasoning and to list the false 
paths he had followed. He lost the sight of his remaining 
eye in 1766 but that scarcely seemed to stop him or even 
slow him down , for he had a phenomenal memory and 
could keep in mind that which would fill several black- 
boards. He published eight hundred papers , some of them 
quite long, and at the time of his death left enough papers 
behind to keep the printing presses busy for thirty-five 
years . 

Euler published a tremendously successful populariza- 
tion of science in 1768, one that remained in print for 
ninety years. He died shortly after working out certain 
mathematical problems in connection with ballooning , 
inspired by the successful flight of the Montgolfier broth- 
ers. He introduced the symbol V' for the base of natural 
logarithms, 'T for the square root of minus one, and 
41 f( )” for functions . 



The Granger Collection 


124 


NUMBERS AND MATHEMATICS 


The Imaginary That Isn't 


125 


Fourth, (-0 4 equals (-i) 2 times (-/)*> which is also 
— 1 times —1, or +1. 

All four suggested solutions, when substituted into the 
equation jc 4 — 1=0, give the expression +1 — 1—0, which 
is correct. 

It might seem all very well to talk about imaginary 
numbers — for a mathematician. As long as some defined 
quantity can be made subject to rules of manipulation that 
do not contradict anything else in the mathematical sys- 
tem, the mathematician is happy. He doesn't really care 
what it “means/' 

Ordinary people do, though, and that’s where my 
sociologist’s charge of mysticism against mathematicians 
arises. 

And yet it is the easiest thing in the world to supply 
the so-called “imaginary” numbers with a perfectly real 
and concrete significance. Just imagine a horizontal line 
crossed by a vertical line and call the point of intersection 
zero. Now you have four lines radiating out at mutual 
right angles from that zero point. You can equate those 
lines with the four kinds of numbers. 

If the line radiating out to the right is marked off at 
equal intervals, the marks can be numbered +1, +2, +3, 
+ 4, . . . , and so on for as long as we wish, if we only 
make the line long enough. Between the markings are all 
the fractions and irrational numbers. In fact, it can be 
shown that to every point on such a line there corresponds 
one and only one positive real number, and for every 
positive real number there is one and only one point on 
the line. 

The line radiating out to the left can be similarly 
marked off with the negative real numbers, so that the 
horizontal line can be considered the “real-number axis,” 
including both positives and negatives. 

Similarly, the line radiating upward can be marked off 
with the positive imaginary numbers, and the one radiat- 
ing downward with the negative imaginary numbers. The 
vertical line is then the imaginary-number axis. 

Suppose we label the different numbers not by the 
usual signs and symbols, but by the directions in which 
the lines point. The rightward line of positive real num- 
bers can be called East because that would be its direction 
of extension on a conventional map. The leftward line of 


negative real numbers would be West; the upward line of 
positive imaginaries would be North; and the downward 
gpe of negative imaginaries would be South. 

Now if we agree that +1 times +1 equals +1, and 
# we concentrate on the compass signs as I have defined 
them, we are saying that East times East equals East. 
Again since —1 times —1 also equals +1, West times 
West equals East. Then, since times +i equals — 1, 
and so does — * times — i, then North times North equals 
West and so does South times South. 

We can also make other combinations such as —1 
times +/, which equals — i (since positive times negative 
yields a negative product even when imaginaries are in- 
volved), so that West times North equals South. If we 
list all the possible combinations as compass points, ab- 
breviating those points by initial letters, we can set up 
the following system: 


ExE-E 
SxE— S 
WxE=W 
NxE=N 


EXS=S 

SxS=W 

WxS=N 

NxS=E 


ExW=W 

SxW^N 

WxW=E 

NXW=S 


ExN=N 

SxN-E 

WxN=S 

NxN=W 


There is a very orderly pattern here. Any compass 
point multiplied by East is left unchanged, so that East 
as a multiplier represents a rotation of 0°. On the other 
hand, any compass point multiplied by West is rotated 
through 180° (“about face”). North and South represent 
right-angle turns. Multiplication by South results in a 
90 fl clockwise turn (“right face”); while multiplication 
by North results in a 90° counterclockwise turn (“left 
face”). 

Now it so happens that an unchanging direction is the 
simplest arrangement, so East (the positive real numbers) 
is easier to handle and more comforting to the soul than 
any of the others. West (the negative real numbers), 
which produces an about face but leaves one on the same 
line at least, is less comforting, but not too bad. North and 
South (the imaginary numbers), which send you off in a 
new direction altogether, are least comfortable. 

But viewed as compass points, you can see that no set 
of numbers is more “imaginary” or, for that matter, more 
“real” than any other. 



126 NUMBERS AND MATHEMATICS 

Now consider how useful the existence of two number 
axes can be. As long as we deal with the real numbers 
only, we can move along the real-number axis, backward 
and forward, one-dimensionally. The same would be true 
if we used only the imaginary-number axis. 

Using both, we can define a point as so far right or 
left on the real-number axis and so far up or down on the 
imaginary-number axis. This will place the point some- 
where in one of the quadrants formed by the two axes. 
This is precisely the manner in which points are located 
on the earth's surface by means of latitude and longitude. 

We can speak of a number such as +5+5 i, which 
would represent the point reached when you marked off 
5 units East followed by 5 units North. Or_you can have 
— 7+6i or +0.5432—9,115/ or +V?+Y 3 *- 

Such numbers, combining real and imaginary units, are 
called “complex numbers.” 

Using both axes, any point in a plane (and not merely 
on a line) can be made to correspond to one and only one 
complex number. Again every conceivable complex num- 
ber can be made to correspond to one and only one point 
on a plane. 

In fact, the real numbers themselves are only special 
cases of the complex numbers, and so, for that matter, 
are the imaginary numbers. If you represent complex 
numbers as all numbers of the form +a+&/, then the 
real numbers are all those complex numbers in which b 
happens to be equal to zero. And imaginary numbers are 
all the complex numbers in which a happens to be equal 
to zero. 

The use of the plane of complex numbers, instead of 
the lines of real numbers only, has been of inestimable 
use to the mathematician. 

For instance, the number of solutions in a polynomial 
equation is equal to its degree only if complex numbers 
are considered as solutions, rather than merely real 
numbers and imaginary numbers. For instance the two 
solutions of * 2 -l=0 are +1 and -1, which can be 
written as +1 + 0/ and -1 + 0/. The two solutions of 
x 2 +l=0 are +/ and — i, or 0+/ and 0—/. The four 
solutions of x 4 — 1=0 are all four complex numbers just 
listed. 


The Imaginary That Isn’t 127 

In all these very simple cases, the complex numbers 
contain zeros and boil down to either real numbers or 
to imaginary numbers. This, nevertheless, is not always 
so. In the equation x 3 — 1— 0 one solution, to be sure, is 
+ 1+0/ {which can be written simply as +1), but the 
other two solutions are —M+lk-yjTi and — Va— 3£y3 /. 

The Gentle Reader with ambition can take the cube 
of either of these expressions (if he remembers how to 
mulitply polynomials algebraically) and satisfy himself 
that it will come out +1. 

Complex numbers are of practical importance too. 
Many familiar measurements involve “scalar quantities” 
which differ only in magnitude. One volume is greater or 
less than another; one weight is greater or less than an- 
other; one density is greater or less than another. For 
that matter, one debt is greater or less than another. For 
all such measurements, the real numbers, either positive 
or negative, suffice. 

However, there are also “vector quantities” which pos- 
sess both magnitude and direction. A velocity may differ 
from another velocity not only in being greater or less, 
but in being in another direction. This holds true for 
forces, accelerations, and so on. 

For such vector quantities, complex numbers are neces- 
sary to the mathematical treatment, since complex num- 
bers include both magnitude and direction (which was my 
reason for making the analogy between the four types of 
numbers and the compass points). 

Now, when my sociology professor demanded “the 
square root of minus one pieces of chalk,” he was speak- 
ing of a scalar phenomenon for which the real numbers 
were sufficient. 

On the other hand, had he asked me how to get from 
his room to a certain spot on the campus, he would 
probably have been angered if I had said, “Go two hun- 
dred yards.” He would have asked, with asperity, “In 
which direction?” 

Now, you see, he would have been dealing with a 
vector quantity for which the real numbers are insufficient. 
I could satisfy him by saying “Go two hundred yards 
northeast,” which is equivalent to saying “Go l00\/2 plus 
100y 2 i yards." 



128 NUMBERS AND MATHEMATICS 

Surely it is as ridiculous to consider the square root of 
minus one "imaginary” because you can’t use it to count 
pieces of chalk as to consider the number 200 as “imagi- 
nary” because by itself it cannot express the location of 
one point with reference to another. 


Pari III 


NUMBERS AND 
MEASUREMENT 



o 

Jr FORGET IT! 


The other day I was looking through a new text- 
book on biology {Biological Science: An Inquiry into 
Life f written by a number of contributing authors and 
published by Harcourt, Brace & World, Inc., in 1963), I 
found it fascinating. 

Unfortunately, though, I read the Foreword first (yes, 
Tm one of that kind) and was instantly plunged into the 
deepest gloom. Let me quote from the first two para- 
graphs; 

“With each new generation our fund of science knowl- 
edge increases fivefold. . . , At the current rate of scienti- 
fic advance, there is about four times as much significant 
biological knowledge today as in 1930, and about sixteen 
times as much as in 1900. By the year 2000, at this rate 
of increase, there will be a hundred times as much biology 
to ‘cover' in the introductory course as at the beginning 
of the century/ 1 

Imagine how this affects me. I am a professional 
“keeper-upper” with science and in my more manic, ebul- 
lient, and carefree moments. I even think I succeed 
fairly well. 

Then I read something like the above-quoted passage 
and the world falls about my ears. I don't keep up with 
science. Worse, I can't keep up with it. Still worse, I f m 
falling farther behind every day. 


131 



132 


NUMBERS AND MEASUREMENT 


And finally, when I’m all through sorrowing for my- 
self I devote a few moments to worrying about the 
world generally. What is going to become of Homo sapiens? 
We’re going to smarten ourselves to death. After a while, 
we will all die of pernicious education, with our brain 
cells crammed to indigestion with facts and concepts, and 
with blasts of information exploding out of our ears. 

But then, as luck would have it, the very day after 1 
read the Foreword to Biological Science I came across 
an old, old book entitled Pike’s Arithmetic. At least that 
is the name on the spine. On the title page it spreads 
itself a bit better, for in those days titles were titles. It 
goes “A New and Complete System of Arithmetic Com- 
posed for the Use of the Citizens of the United States, 
by Nicolas Pike, A.M.” It was first published in 1785, 
but the copy I have is only the 11 ‘Second Edition, En- 

larged,” published in 1797. , 

It is a large book of over 500 pages, crammed full of 
small print and with no relief whatever in the way of 
illustrations or diagrams. It is a solid slab of arithmetic 
except for small sections at the very end that introduce 

algebra and geometry. . . . . , 

I was amazed. I have two children in grade school 
(and once I was in grade school myself), and I know 
what arithmetic books are like these days. They are 
nowhere near as large. They can’t possibly have even one 
fifth the wordage of Pike. 

Can it be that we are leaving anything out? 

So I went through Pike and, you know, we are leaving 
something out. And there’s nothing wrong with that. The 
trouble is we’re not leaving enough out. 

On page 19, for instance, Pike devotes half a page to 
a listing of numbers as expressed in Roman numerals, 
extending the list to numbers as high as five hundred 

thousand. . . 

Now Arabic numerals reached Europe in the High 
Middle Ages, and once they came on the scene tne 
Roman numerals were completely outmoded (see Chapter 
1). They lost all possible use, so infinitely superior was 
the new Arabic notation. Until then who knows how many 

[* This article first appeared in March 1964, and time marches on. My 
younger child is now well along In ccftege.] 


Forget it! 


133 


reams of paper were required to explain methods for 
calculating with Roman numerals. Afterward the same 
calculations could be performed with a hundredth of the 
explanation. No knowledge was lost — only inefficient rules. 

And yet five hundred years after the deserved death of 
the Roman numerals, Pike still included them and ex- 
pected his readers to be able to translate them into Arabic 
numerals and vice versa even though he gave no instruc- 
tions for how to manipulate them. In fact, nearly two 
hundred years after Pike, the Roman numerals are still 
being taught. My little daughter is learning them now. 

But why? Where's the need? To be sure, you will find 
Roman numerals on cornerstones and gravestones, on 
clockfaces and on some public buildings and documents, 
but it isn't used for any need at all. It is used for show, 
for status, for antique flavor, for a craving for some kind 
of phony classicism. 

I dare say there are some sentimental fellows who feel 
that knowledge of the Roman numerals is a kind of gate- 
way to history and culture; that scrapping them would be 
like knocking over what is left of the Parthenon, but I 
have no patience with such mawkishness. We might as 
well suggest that everyone who learns to drive a car be 
required to spend some time at the wheel of a Model-T 
Ford so he could get the flavor of early cardom. 

Roman numerals? Forget it! — And make room in- 
stead for new and valuable material. 

But do we dare forget things? Why not? We've for- 
gotten much; more than you imagine. Our troubles stem 
not from the fact that we've forgotten, but that we re- 
member too well; we don't forget enough. 

A great deal of Pike’s book consists of material we 
have imperfectly forgotten. That is why the modern 
arithmetic book is shorter than Pike. And if we could 
but perfectly forget, the modern arithmetic book could 
grow still shorter. 

For instance. Pike devotes many pages to tables — pre- 
sumably important tables that he thought the reader ought 
to be familiar with. His fifth table is labeled “cloth 
measure.” 

Did you know that 2V4 inches make a “nail”? Well, 
they do. And 16 nails make a yard; while 12 nails make 
an ell. 

No, wait awhile. Those 12 nails (27 inches) make a 



134 


numbers and measurement 


i »-* «u. i. uj. » r&'enS 

<37* inches) ™ke . 

" >-■« t *a 

^rel“-r£y"c» figure some way of gening .he 

ell out of business o{ goo d s is measured 

Furthermore, almost ev > P^ 8 f butter> a punc h 

in its owumts. Yo^Pjd ^ stone of butcher’s meat, 

IdTot. Each 1 of these SSX” 

- - 0 . - 

Pike carefully gives all the ®^ u ' 0 Well, how about 

Do you want to 1 pole; 

8 sirs' ss 

mon line of work in Jr° H 2 pints make a quart 

gallon. Well, we still know .ha. 

* tg$Ei .imes. however, a "«££" v *'5 Z 

ale was hut a sWr £ r ; f ™* t JSJd quanttlies. Well, 8 gal- 
know how to speak of m kes “ a fi r kin of ale in 

Ions made a firkin— that is, „ t ma ke “a firkin 

London.” » ^ ^‘^i^rSdiate quantity, 8% gal- 
of beer m London. Ihe jniermc u ^ beer ”_ pre - 

lons, is marked down as ^ * q£ London where the 
povtirchfr »e,e less finicky in dis.i.guish.ng 

h ST^= 2 ^’aSKn'^SSS 

‘ ™ "i-h SS 1 hogshead; 2 bar- 

Tef make a puncheon; and 3 barrels make a butt. 

in case your appetite has 
been sharpened for something sl ‘'l make a pottle. 

^fhMn SSS »' 

of a*pottle!) But let's proceed. 


Forget It! 


135 


Next, 2 pottles make a gallon, 2 gallons make a peck, 
and 4 pecks make a bushel. (Long breath now.) Then 
2 bushels make a strike, 2 strikes make a coom, 2 cooms 
make a quarter, 4 quarters make a chaldron (though in 
the demanding city of London, it takes 414 quarters to 
make a chaldron). Finally, 5 quarters make a wey and 
2 weys make a last. 

Fm not making this up. I’m copying it right out of 
Pike, page 48. 

Were people who were studying arithmetic in 1797 
expected to memorize all this? Apparently, yes, because 
Pike spends a lot of time on compound addition. That’s 
right, compound addition. 

You see, the addition you consider addition is just 
“simple addition.” Compound addition is something 
stronger and I will now explain it to you. 

Suppose you have 15 apples, your friend has 17 apples, 
and a passing stranger has 19 apples and you decide to 
make a pile of them. Having done so, you wonder how 
many you have altogether. Preferring not to count, you 
draw upon your college education and prepare to add 
15+17+19. You begin with the units column and find 
that 5+7+9=21. You therefore divide 21 by 10 and 
find the quotient is 2 plus a remainder of 1, so you put 
down the remainder, 1, and carry the quotient 2 into 
the tens col — 

I seem to hear loud yells from the audience. ‘‘What is 
all this?” comes the fevered demand. “Where does this 
‘divide by 10’ jazz come from?” 

Ah, Gentle Readers, but this is exactly what you do 
whenever you add. It is only that the kindly souls who 
devised our Arabic system of numeration based it on the 
number 10 in such a way that when any two-digit num- 
ber is divided by 10, the first digit represents the quotient 
and the second the remainder. 

For that reason, having the quotient and remainder in 
our hands without dividing, we can add automatically. 
If the units column adds up to 21, we put down 1 and 
carry 2; if it had added up to 57, we would have put 
down 7 and carried 5, and so on. 

The only reason this works, mind you, is that in adding 
a set of figures, each column of digits (starting from the 
right and working leftward) represents a value ten times 



136 NUMBERS AND MEASUREMENT 


THE YARDSTICK 

The yardstick is one of those accompaniments of life that 
we tend to take for granted. Few people have any idea 
how difficult it was to produce one and how many subtle 
concepts had to be embraced before the yardstick became 
possible. 

The natural manner of measuring lengths in early times 
was to use various portions of the body for the purpose . 
We still talk of “hands" in measuring the heights of horses, 
and of a "span" for the length represented by the out- 
stretched fingers. A “cubit" from a Latin word meaning 
“elbow” is the distance from fingertips to elbow , and a 
“yard” (related to “girth”) is the distance from fingertips 
to nose t or the waist measurement of a man . 

The trouble with using portions of the body as a meas- 
uring device is that the lengths and measurements of those 
portions varies from person to person. The length from 
my fingertip to my nose is quite close to a yard t but my 
waist measurement is distinctly greater than a yard . 

It finally occurred to people to establish a "standard 
yard”and never mind what your own measurements are. 
According to tradition, a standard yard was originally 
adjusted to the length from the fingertips of King Henry 
I of England to his nose . (And the standard foot is sup- 
posed to be based on the foot of Charlemagne.) 

Naturally, the King of England can't travel from vil- 
lage to village measuring out lengths of cloth from his 
nose to his fingertips. Instead a stick was held up against 
him and marks are made at his nose and his fingertips. 
The distance between the marks is a standard yard. Other 
sticks can be measured off against that standard and can 
become secondary standards to be sent to every village 
for use in checking the activities of the local merchants. 



The Granger Collection 




138 


NUMBERS AND MEASUREMENT 


Forget It! 


139 


as great as the column before. The rightmost column is 
units, the one to its left is tens, the one to its left is hun- 
dreds, and so on. 

It is this combination of a number system based on ten 
and a value ratio from column to column of ten that 
makes addition very simple. It is for this reason that it 
is, as Pike calls it, ‘'simple addition/’ 

Now suppose you have 1 dozen and 8 apples, your 

friend has 1 dozen and 10 apples, and a passing stranger 
has 1 dozen and 9 apples. Make a pile of those and add 
them as follows: 

1 dozen 8 units 

1 dozen 10 units 

1 dozen 9 units 

Since 8 + 10+9“27, do we put down 7 and carry 2? 
Not at all? The ratio of the “dozens” column to the j 
“units” column is not 10 but 12, since there are 12 units ■ 
to a dozen. And since the number system we are using 
is based on 10 and not on 12, we can no longer let the 
digits do our thinking for us. We have to go long way 
round. 

If 8 + 10+9^27, we must divide that sum by the ratio 
of the value of the columns; in this case, 12. We find that 
27 divided by 12 gives a quotient of 2 plus a remainder 
of 3, so we put down 3 and carry 2. In the dozens column | 
we get 1 + 1 + 1 + 2 — 5. Our total therefore is 5 dozen \ 
and 3 apples. 

Whenever a ratio of other than 10 is used so that you 
have to make actual divisions in adding, you have “com- 
pound addition.” You must indulge in compound addi- 
tion if you try to add 5 pounds 12 ounces and 6 pounds 
8 ounces, for there are 16 ounces to a pound. You are 
stuck again if you add 3 yards 2 feet 6 inches to 1 yard 

2 feet 8 inches, for there are 12 inches to a foot, and 

3 feet to a yard, 

You do the former if you care to; 111 do the latter. 
First, 6 inches and 8 inches are 14 inches. Divide 14 by 
12, getting 1 and a remainder of 2, so you put down 2 
and carry 1. As for the feet, 2+2+1 -5. Divide 5 by 3 
and get I and a remainder of 2, put down 2 and carry 
1. In the yards, you have 3 + 1 + 1=5. Your answer, then, 
is 5 yards 2 feet 2 inches. 


Now why on Earth should our unit ratios vary all over 
the lot, when our number system is so firmly based on 10? 
There are many reasons (valid in their time) for the use 
of odd ratios like 2, 3, 4, 8, 12, 16, and 20, but surely 
we are now advanced and sophisticated enough to use 10 
as the exclusive (or nearly exclusive) ratio. If we could 
do so, we could with much pleasure forget about com- 
pound addition— and compound subtraction, compound 
multiplication, compound division, too. (They also exist, 
of course.) 

To be sure, there are times when nature makes the 
universal ten impossible. In measuring time, the day and 
the year have their lengths fixed for us by astronomical 
conditions and neither unit of time can be abandoned. 
Compound addition and the rest will have to be retained 
for such special cases, alas. 

But who in blazes says we must measure things in firkins 
and pottles and Flemish ells? These are purely manmade 
measurements, and we must remember that measures 
were made for man and not man for measures. 

It so happens that there is a system of measurement 
based exclusively on ten in this world. It is called the 
metric system and it is used all over the civilized world 
except for certain English-speaking nations such as the 
United States and Great Britain. 

By not adopting the metric system, we waste our time 
for we gain nothing, not one thing, by learning our own 
measurements. The loss of time (which is expensive 
indeed) is balanced by not one thing I can imagine. (To 
be sure, it would be expensive to convert existing instru- 
ments and tools but it would have been nowhere nearly 
as expensive if we had done it a century ago, as we 
should have.) 

There are those, of course, who object to violating our 
long-used cherished measures. They have given up cooms 
and chaldrons but imagine there is something about inches 
and feet and pints and quarts and pecks and bushels that 
is “simpler” or “more natural” than meters and liters. 

There may even be people who find something dan- 
gerously foregn and radical (oh, for that vanished word 
of opprobrium, “Jacobin”) in the metric system — yet it 
was the United States that led the way. 

In 1786, thirteen years before the wicked French revo- 
lutionaries designed the metric system, Thomas Jefferson 



140 NUMBERS AND MEASUREMENT 

(a notorious “Jacobin” according to the Federalists, at 
least) saw a suggestion of his adopted by the infant 
United States. The nation established a decimal currency. 

What we had been using was British currency, and that 
is a fearsome and wonderful thing. Just to point out how 
preposterous it is. let me say that the British people who, 
over the centuries, have, with monumental patience, 
taught themselves to endure anything at all provided it 
was “traditional”- — are now sick and tired of their cur- 
rency and are debating converting it to the decimal sys- 
tem. (They can’t agree on the exact details of the 

change.)* . . , _ 

But consider the British currency as it has been, io 
begin with, 4 farthings make 1 penny; 12 pennies make 
1 shilling, and 20 shillings make 1 pound. In addition, 
there is a virtual farrago of terms, if not always actual 
coins, such as ha’pennies and thruppences and sixpences 
and crowns and half-crowns and florins and guineas and 
heaven knows what other devices with which to cripple 
the mental development of the British schoolchild and 
line the pockets of British tradesmen whenever tourists 
come to call and attempt to cope with the currency. 

Needless to say, Pike gives careful instruction on how 
to manipulate pounds, shillings, and pence and very 
special instructions they are. Try dividing 5 pounds, 13 
shillings, 7 pence by 3. Quick now! 

In the United States, the money system, as originally 
established, is as follows: 10 mills make 1 cent; 10 cents 
make 1 dime; 10 dimes make 1 dollar; 10 dollars make 
1 eagle. Actually, modern Americans, in their calculations, 
stick to dollars and cents only. 

The result? American money can be expressed in deci- 
mal form and can be treated as can any other decimals. 
An American child who has learned decimals need only 
be taught to recognize the dollar sign and he is all set. 
In the time that he does, a British child has barely mas- 
tered the fact that thruppence ha’penny equals 14 farthings. 

What a pity that when, thirteen years later, in 1799, 
the metric system came into being, our original anti- 

Since this article wzs written, the British have carried through the 
change. When I visited Great Britain in 1974 f I was greatly disappointed 
at not being able to deal with threepences and half crowns. They are 
also adopting the metric system , leaving the intransigent United States 
virtually atone in opposition .] 


Forget It I 141 

British* pro-French feelings had not lasted just long 
enough to allow us to adopt it. Had we done so, we would 
have been as happy to forget our foolish pecks and 
ounces, as we are now happy to have forgotten our pence 
and shillings. (After all, would you like to go back to 
British currency in preference to our own?) 

What I would like to see is one form of money do for 
all the world. Everywhere. Why not? 

I appreciate the fact that I may be accused because of 
this of wanting to pour humanity into a mold, and of 
being a conformist. Of course, I am not a conformist 
(heavens!). I have no objection to local customs and local 
dialects and local dietaries. In fact, I insist on them for I 
constitute a locality all by myself. I just don’t want to 
keep provincialisms that were well enough in their time 
but that interfere with human well-being in a world 
which is now 90 minutes in circumference. 

If you think provincialism is cute and gives humanity 
color and charm, let me quote to you once more from 
Pike. 

“Federal Money” (dollars and cents) had been intro- 
duced eleven years before Pike’s second edition, and he 
gives the exact wording of the law that established it and 
discusses it in detail — under the decimal system and not 
under compound addition. 

Naturally, since other systems than the Federal were 
still in us, rules had to be formulated and given for con- 
verting (or “reducing”) one system to another. Here is 
the list. I won’t give you the actual rules, just the list of 
reductions that were necessary, exactly as he lists them: 

L To reduce New Hampshire, Massachusetts, Rhode 
Island, Connecticut, and Virginia currency: 

1 . To Federal Money 

2. To New York and North Carolina currency 

3. To Pennsylvania, New Jersey, Delaware, and 

Maryland currency 

4. To South Carolina and Georgia currency 

5. To English money 

6. To Irish money 

7. To Canada and Nova Scotia currency 

8. To Livres Tournois (French money) 

9. To Spanish milled dollars 



142 


NUMBERS AND MEASUREMENT 


Forget It! 


143 


II. To reduce Federal Money to New England and 
Virginia currency. 

HI. To reduce New Jersey, Pennsylvania, Delaware, 
and Maryland currency: 

1. To New Hampshire, Massachusetts, Rhode 

Island, Connecticut, and Virginia currency. 

2. To New York and . - , 

Oh, the heck with it. You get the idea. 

Can anyone possible be sorry that all that cute pro- 
vincial flavor has vanished? Are you sorry that every 
time you travel out of state you don’t have to throw 
yourself into fits of arithmetical discomfort whenever you 
want to make a purchase? Or into similar fits every time 
someone from another state invades yours and tries to 
dicker with you? What a pleasure to have forgotten all 
that. 

Then tell me what’s so wonderful about having fifty 
sets of marriage and divorce laws? 

In 1752, Great Britain and her colonies (some two 
centuries later than Catholic Europe) abandoned the 
Julian calendar and adopted the astronomically more 
correct Gregorian calendar (see Chapter 11). Nearly half 
a century later, Pike was still giving rules for solving com- 
plex calendar-based problems for the Julian calendar as 
well as for the Gregorian. Isn’t it nice to have forgotten 
the Julian calendar? 

Wouldn’t it be nice if we could forget most of calen- 
drical complications by adopting a rational calendar that 
would tie the day of the month firmly to the day of the 
week and have a single three-month calendar serve as a 
perpetual one, repeating itself over and over every three 
months? There is a world calendar proposed which would 
do just this. 

It would enable us to do a lot of useful forgetting. 

I would like to see the English language come into 
worldwide use. Not necessarily as the only language or 
even as the major language. It would just be nice if every- 
one — whatever his own language was — could also speak 
English fluently. It would help in communications and 
perhaps, eventually, everyone would just choose to speak 
English. 


That would save a lot of room for other things. 

Why English? Well, for one thing more people speak 
English as either first or second language than any other 
language on Earth, so wc have a head start. Secondly, 
far more science is reported in English than in any other 
language and it is communication in science that is critical 
today and will be even more critical tomorrow. 

To be sure, we ought to make it as easy as possible for 
people to speak English, which means we should rational- 
ize its spelling and grammar. 

English, as it is spelled today, is almost a set of Chinese 
ideograms. No one can be sure how a word is pronounced 
by looking at the letters that make it up. How do you 
pronounce: rough, through, though, cough, hiccough, and 
lough; and why is it so terribly necessary to spell all those 
sounds with the mad letter combination “ough”? 

It looks funny, perhaps, to spell the words ruff, throo, 
thoh, cawf, hiccup, and lokh; but we already write hiccup 
and it doesn’t look funny. We spell colour, color, and 
centre, center, and shew, show and grey, gray. The result 
looks funny to a Britisher but we are used to it. We can 
get used to the rest, too, and save a lot of wear and tear 
on the brain. We would all become more intelligent, if 
intelligence is measured by proficiency at spelling, and 
we’ll not have lost one thing. 

And grammar? Who needs the eternal hair-splitting 
arguments about “ ‘shall” and “will” or “which” and 
“that”? The uselessness of it can be demonstrated by the 
fact that virtually no one gets it straight anyway. Aside 
from losing valuable time, blunting a child’s reasoning 
faculties, and instilling him or her with a ravening dislike 
for the English language, what do we gain? 

If there be some who think that such blurring of fine 
distinctions will ruin the language, I would like to point 
out that English, before the grammarians got hold of it, 
had managed to lose its gender and its declensions almost 
everywhere except among the pronouns. The fact that we 
have only one definite article (the) for all genders and 
cases and times instead of three, as in French ( le , la, les) 
or six, as in German ( der t die , das } dem t den , des) in no 
way blunts the English language, which remains an 
admirably flexible instrument. We cherish our follies only 
because we are used to them, and not because they are 
not really follies. 



144 


NUMBERS AND MEASUREMENT 


AMERICAN BILLS 

It is rather heartbreaking that the United States , having 
started resolutely in the right direction, did not continue. 

Immediately after the Revelottiionary War , anti-British 
feeling was strong enough to cause many Americans to 
want to do away with anything trivial that would remind 
them of the hated foe. The “rights of Englishmen" were 
not trivial, so those were congealed into the Bill of Rights. 
The system of coinage , however familiar T was trivial 

The key person in this respect was a Pennsylvanian, 
Gouverneur Morris. He had been a Federalist, an advocate 
of a strong central government over the quarreling and 
disunited states who made up what was wrongly called 
the “United” States immediately after the Revolution. He 
was a member of the Constitutional Congress and , more 
than anyone else, is responsible for the actual wording 
of the Constitution and the casting of it into a clear and 
simple phraseology , devoid of fustian and melodrama. 

It was he, also , who suggested that the United States 
adopt a new coinage based on a decimal system « The 
basic unit, the “ dollar ” ( the paper form of which is pre- 
sented in the illustration, though it scarcely needs it, so 
familiar is it to all of us), gets its name the long way 
round . Back about 1500 , silver from the silver mines in 
Joachim* s Valley (in what is now northwestern Czecho- 
slovakia) was used to coin ounce-pieces. Joachim T s Val- 
ley is, in German, Joachimsthal, and the coins were 
called “Joachminsthalers” or, for short , “ thalers ” or, in 
English, “dollars” 

In colonial times, Spanish coins of about the value of 
the well-known dollars existed. The Spaniards called 
them “pesos 1 ? the English “dollars" and the Americans 
adopted the name and began coining them in 1794. 



The Granger Collection 




146 NUMBERS AND MEASUREMENT 

We must make room for expanding knowledge, or at 
least make as much room as possible. Surely it is as 
important to forget the old and useless as it is to learn 
the new and important. 

Forget it, I say, forget it more and more. Forget it! 

But why am I getting so excited? No one is listening to 
a word I say. 



PRE-FIXING 
IT UP 


I go through life supported and bolstered by many 
comforting myths, as do all of us. One of my own par- 
ticularly cherished articles of faith is that there are no 
arguments against the metric system and that the common 
units make up an indefensible farrago of nonsense that 
we keep only out of stubborn folly. 

Imagine the sobering effect, then, of having recently 
come across a letter by a British gentleman who bitterly 
denounced the metric system as being artificial, sterile, 
and not geared to human needs. For instance, he said 
(and I don’t quote exactly), if one wants to drink beer, 
a pint of beer is the thing. A liter of beer is too much 
and half a liter is too little, but a pint, ah, that’s just right.* 
As far as I can tell, the gentleman was serious in his 
provincialism, and in considering that that to which he 
is accustomed has the force of a natural law. It reminds 
me of the pious woman who set her face firmly against 
all foreign languages by holding up her Bible and saying, 
“If the English language was good enough for the prophet 
Isaiah, and the apostle Paul, it is good enough for me.” 


* Before you write to tell me that half a liter is larger than a pint , let 
me explain that though it is larger than an American pint , if is smaller 
than a British pint , 


147 



148 NUMBERS AND MEASUREMENT 

But mainly it reminds me that 1 want to write an essay 
on the metric system. 

In order to do so, I want to begin by explaining that 
the value of the system does not lie in the actual size of 
the basic units. Its worth is this: that it is a logical system. 
The units are sensibly interrelated. 

All other sets of measurements with which I am ac- 
quainted use separate names for each unit involving a 
particular type of quantity. In distance, we ourselves have 
miles, feet, inches, rods, furlongs, and so on. In volume, 
we have pecks, bushels, pints, drams. In weight, we have 
. ounces, pounds, tons, grains. It is like the Eskimos, who 
are supposed to have I don’t know how many dozens of 
words for snow, a different word for it when it is falling 
or when it is lying there, when it is loose or packed, wet 
or dry, new-fallen or old-fallen, and so on. 

We ourselves see the advantage in using adjective-noun 
combinations. We then have the noun as a genera! term 
for all kinds of snow and the adjective describing the 
specific variety: wet snow, dry snow, hard snow, soft snow, 
and so on. What’s the advantage? First, we see a generali- 
zation we did not see before. Second, we can use the same 
adjectives for other nouns, so that we can have hard rock, 
hard bread, hard heart, and consequently see a new 
generalization, that of hardness. 

The metric system is the only system of measurement 
which, to my knowledge, has advanced to this stage. 

Begin with an arbitrary measure of length, the meter 
(from the Latin metrum or the Greek metron , both mean- 
ing “to measure”). Leave that as the generic term for 
length, so that all units of length are meters. Differentiate 
one unit of length from another by means of an adjective. 
That in my opinion, would be fixing it up right. 

To be sure, the adjectives in the metric system (lest they 
get lost by accident, I suppose) are firmly jointed to the 
generic word and thus become prefixes. (Yes, Gentle 
Reader, in doing this to the measurement system, they 
were “pre-fixing it up.”) 

The prefixes were obtained out of Greek and Latin in 
accordance with the following little table: 

ENGLISH GREEK LATIN 

thousand chilioi mille 

hundred hecaton centum 

ten deka decern 


Pre-Fixing It Up 149 

Now, if we save the Greek for the large units and the 
I^tin for the small ones, we have : 


1 kilometer * 

equals 

1000 

meters 

1 hectometer 

equals 

100 

meters 

1 dekameter 

equals 

10 

meters 

1 meter 

equals 

1 

meter 

1 decimeter 

equals 

0.1 

meter 

1 centimeter 

equals 

0.01 

meter 

1 millimeter 

equals 

0.001 

meter 

It doesn’t matter 

how long 

a meter is; all 

the other 

units of length are as defined. If you happen to 

know the 


length of the meter in terms of yards or of wavelengths 
of light or of two marks on a stick, you automatically 
know the lengths of all the other units. Furthermore, by 
having all the sub-units vary by powers of ten, it becomes 
very easy (given our decimal number system) to convert 
one into another. For instance, I can tell you right off 
that there are exactly one million millimeters in a kilo- 
meter. Now you tell me right off how many inches there 
are in a mile. 

And again, once you have the prefixes memorized, they 
will do for any type of measurement. If you are told that 
a “poise” is a measure of viscosity, it doesn’t matter how 
large a unit it is or how it is related to other sorts of 
units or even what, exactly, viscosity is. Without knowing 
anything at all about it, you still know that a centipoise 
is equal to a hundredth of a poise, that a hectare is a 
hundred ares, that a decibel is a tenth of a bel; and even 
that a “kilobuck” is equal to a thousand dollars.** 

In one respect and, to my mind, in only one were the 
French scientists who established the metric system in 
1795 shortsighted. They did not go past the thousand 
mark in their prefix system. 


* The Greek ch has the guttural German ch sound . The French, who 
invented the metric system, have no such sound in their language and 
used k instead as the nearest approach . That is why chilioi becomes kilo. 
Since we don’t have the guttural ch either, this suits us fine. 

** If anyone wants to write that a millipede is a thousandth of a pede and 
that one centipede equals ten millipedes, by all means, do — but I won’t 
listen . 



150 


NUMBERS AND MEASUREMENT 


Pre-Fixing It Up 


151 


Perhaps they felt that once a convenient basic unit was 
selected for some measurable quantity, then a sub-unit a 
thousand times larger would be the largest useful one, 
while a sub-unit a thousandth as large would be the small- 
est, Or perhaps they were influenced by the fact that there 
is no single word in Latin for any number higher than a 
thousand. (Words like million and billion were invented 
in the late middle ages and in early modern times.) 

The later Greeks, to be sure, used myrias for ten thou- 
sand, so it is possible to say “myriameter” for ten thou- 
sand meters, but this is hardly ever used. People say “ten 
kilometers" instead. 

The net result, then, is that the metric system as or- 
ganized originally offers prefixes that cover only six orders 
of magnitude. The largest unit, “kilo," is one million 
(10*) times as great as the smallest unit “nrilli," and it is 
the exponent, 6, that marks the orders of magnitude. 

Scientists could not, however, stand still for this. Six 
orders of magnitude may do for everyday life, but as the 
advance of instrumentation carried science into the very 
large and very small in almost every field of measure- 
ment, the system simply had to stretch. 

Unofficial prefixes came into use for units above the 
kilo and below the milli and of course that meant the 
danger of nonconformity (which is a bad thing in scientific 
language). For instance, what we call a “Bev” (billion 
electron-volts), the British call a “Gev" ( giga-electron- 
volts). 

In 1958, then, an extended set of prefixes, at intervals 
of three orders of magnitude, was agreed upon by the 
International Committee on Weights and Measures in 
Paris. Here they are, with a couple of the older ones 
thrown in for continuity: 


SIZE 

PREFIX 

GREEK ROOT 

trillion (10 12 ) 

tera- 

ter as (“monster") 

billion (10*) 

giga- 

gigas (“giant") 

million (10 5 ) 

mega- 

megas (“great”) 

thousand (IQ 3 ) 

kilo- 


one (10°) 



thousandth (10 -3 ) 

milli- 


millionth (10 e ) 

micro- 

mikros (“small") 

billionth (10 *) 

nano- 

nanos (“dwarf”) 

trillionth (10 12 ) 

pico- 



The prefix pico- does not have a Greek root. 

Well, then, we have a “picometer" as a trillionth of a 
meter, a “nanogram" as a billionth of a gram, a “giga- 
second" as a billion seconds, and a “teradyne" as a trillion 
dynes. Since the largest unit, the ter a, is 10 24 times the 
smallest unit, the pico, the metric system now stretches 
not merely over 6, but over a full 24 orders of magnitude. 

In 1962 femto- was added for a quadrillion^ (1CT 15 ) 
and atto- for a quintillionth (10 _]8 ). Neither prefix has a 
Greek root.* This extends the metric system over 30 
orders of magnitude. 

Is this too much? Have we overdone it, perhaps? Well, 
let’s see. 

The metric unit of length is the meter. I won’t go into 
the story of how it was fixed at its precise length, but that 
precise length in terms of familiar units is 1.093611 
yards or 39.37 inches. 

A kilometer, naturally, is a thousand times that, or 
1093.6 yards, which comes out to 0.62137 mile. We 
won’t be far off if we call a kilometer % of a mile. A 
mile is sometimes said to equal “twenty city blocks"; 
that is, the distance between, let us say, 59th Street and 
79th Street in Manhattan. If so. a kilometer would repre- 
sent 12}£ city blocks, or the distance from halfway be- 
tween 66th and 67th streets to 79th Street. 

For a megameter we increase matters three orders of 
magnitude and it is equal to 621.37 miles. This is a con- 
venient unit for planetary measurements. The air distance 
from Boston, Massachusetts, to San Francisco, California, 
is just about 4H megameters. The diameter of the earth 
is 12?4 megameters and the circumference of the earth is 
about 40 megameters. And finally, the moon is 380 mega- 
meters from the earth. 

Passing on to the gigameter, we have a unit 621,370 
miles long, and this comes in handy for the nearer portions 
of the solar system. Venus at its closest is 42 gigameters 
away and Mars can approach us as closely as 58 giga- 
meters. The sun is 145 gigameters from the earth and 
Jupiter, at its closest, is 640 gigameters distant; at its 
farthest, 930 gigameters away. 


[* / did not give the non-Creek roots when this article first appeared in 
November 1962, but I will now. Pico is from the Spanish word for 
ts small Femto and atto are from the Danish words for "fifteen" and 
"eighteen" respectively. ] 



152 


NUMBERS AND MEASUREMENT 


Pre-Fixing It Up 


153 


ANDROMEDA GALAXY 


The Andromeda Galaxy , mentioned briefly in this article , 
has one unusual distinction. It is the farthest object that 
can be seen with the unaided eye — so if anyone asks you 
how far you can see ( with glasses on, if you're near- 
sighted), tell him 2,300,000 light-years. 

The Andromeda looks like a faint, fuzzy object of about 
the fourth magnitude . // is not likely to be noticed by a 
casual sky -gazer, but it was noted in the star maps of 
some of the Arab astronomers of the Middle Ages. The 
first to describe it among our Western astronomers was 
the German observer Simon Marius, in 1612 . 

In the next century, a French observer, Charles Messier , 
was interested in recording all the permanently fuzzy 
objects in the sky so that they not be mistaken for comets. 
(Messier was interested in comets.) The Andromeda was 
thirty- first on his list, and its alternate name, still often 
used, is M3L 

In the simple telescopes of the 1700s, the Andromeda 
looked like a whirling cloud of gas, and the French 
astronomer Pierre Simon de Laplace thought that was 
indeed what it was. In a popular book on astronomy he 
wrote in the early 1800s, he made the suggestion in an 
appendix. Stars like our Sun and the planets that accom- 
pany them originated out of a whirling, condensing cloud 
of gas like that of the Andromeda . The Andromeda was 
then called Andromeda Nebula (from the Latin word 
for “cloud”), and Laplace's suggestion has always been 
called the rt nebular hypothesis” 

In recent years a vastly more sophisticated form of the 
nebular hypothesis has come to be accepted as the origin 
of the solar system, but the Andromeda is no cloud of 
gas. It is a collection of stars as large as, or larger than, 
our own Milky Way Galaxy , and farther beyond are 
billions of other galaxies. 



The Granger Collection 



154 


NUMBERS AND MEASUREMENT 


Pre- Fixing It Up 


155 


Finally, by stretching to the limit of the newly ex- 
tended metric system, we have the terameter, equal to 
621,370,000 miles. This will allow us to embrace the 
entire solar system. The extreme width of Pluto's orbit, 
for instance, is not quite 12 terameters. 

The solar system, however, is just a speck in the 
Galaxy. For measuring distances to the stars, the two most 
common units are the light-year and the parsec, and both 
are outside the metric system. What's more, even the new 
extension of the system can't reach them. The light-year 
is the distance that light travels in one year. This is about 
5,880,000,000,000 miles or 9450 terameters. The parsec 
is the distance at which a star would appear to us to have 
a parallax of one second of arc (parallax-second, get it), 
and that is equal to 3.26 light-years, or about 30,000 
terameters. 

Even these nonmetric units err on the small side. If one 
were to draw a sphere about the solar system with a 
radius of one parsec, not a single known star would be 
found within that sphere. The nearest stars, those of the 
Alpha Centauri system, are about 1.3 parsecs away. 
There are only thirty-three stars, out of a hundred billion 
or so in the Galaxy, closer to our sun than four parsecs, 
and of these only seven are visible to the naked eye. 

There are many stars beyond this — far beyond this. 
The Galaxy as a whole has a diameter which is, at its 
longest, 30,000 parsecs. Of course, we might use the 
metric prefixes and say that the diameter of the Galaxy 
is 30 kiloparsecs. 

But then the Galaxy is only a speck in the entire uni- 
verse. The nearest extragalactic structures are the Magel- 
lanic Clouds, which are 50 kiloparsecs away, while the 
nearest full-size galaxy to our own is Andromeda, which 
is 700 kiloparsecs away. And there are hundreds of bil- 
lions of galaxies beyond at a distance of many mega- 
parsecs. 

The farthest galaxies that have been made out have 
distances estimated at about two billion parsecs, which 
would mean that the entire visible universe, as of now, 
has a diameter of about 4 gigaparsecs.* 


[• Since this article was written J quasars have been detected at distances 
of 4 gigaparsecs so the visible universe has a diameter of 8 gigaparsecs. ] 


Suppose, now, we consider the units of length in the 
other direction — toward the very small. 

A micrometer is a good unit of length for objects 
visible under the ordinary optical microscope. The body 
cells, for instance, average about 4 micrometers in diam- 
eter. (A micrometer is often called a “micron,”) 

Drop down to the nanometer (often called a “milli- 
micron”) and it can be conveniently used to measure 
the wavelengths of visible light. The wavelength of the 
longest red light is 760 nanometers, while that of the 
shortest violet light is 380 nanometers, Ultraviolet light 
has a range of wavelengths from 380 nanometers down to 
1 nanometer. 

Shrinking the metric system still further, we have the 
picometer, or a trillionth of a meter. Individual atoms 
have diameters of from 100 to 600 picometers. And soft 
gamma rays have wavelengths of about 1 picometer. 

The diameter of subatomic particles and the wave- 
lengths of the hard gamma rays go well below the pico- 
meter level, however, reaching something like 1 femto- 
meter. 

The full range of lengths encountered by present-day 
science, from the diameter of the known universe at one 
extreme, to the diameter of a subatomic particle at the 
other, covers a range of 41 orders of magnitude. In other 
words, it would take 10 11 protons laid side by side to 
stretch across the known universe. 

What about mass? 

The fundamental unit of mass in the metric system is 
the gram, a word derived from the Greek gramma , mean- 
ing a letter of the alphabet.* It is a small unit of weight, 
equivalent to ^§ 35 ounces. A kilogram, or a thousand 
grams, is equal to 2.205 pounds, and a megagram is 
therefore equal to 2205 pounds. 

The megagram is almost equal to the long ton (2240 
pounds) in our own units, so it is sometimes called the 
“metric ton” or the “tonne.” The latter gives it the French 
spelling, but doesn't do much in the way of differentiating 
the pronunciation, so I prefer metric ton. 


* The Greeks marked smalt weights with letters of the alphabet to indi- 
cate their weight J for they used letters to represent numbers, too. 



156 NUMBERS AND MEASUREMENT 

A gigagram is 1000 metric tons and a teragram is 

1.000. 000 metric tons and this is large enough by com- 
mercial standards. These don’t even begin, however, to 
scratch the surface astronomically. Even a comparatively 
small body like the moon has a mass equal to 73 trillion 
teragrams, The earth is 81 times more massive and has a 
mass of nearly 6 quadrillion teragrams. And the sun, a 
merely average star, has a mass 330,000 times that of the 
earth. 

Of course, we might use the sun itself as a unit of 
weight. For instance the Galaxy has a total mass equal to 

150.000. 000.000 times that of the sun, and we could 
therefore say that the mass of the Galaxy is equal to 150 
gigasuns. Since it is also estimated that in the known 
universe there are at least 100,000,000,000 galaxies, then, 
assuming ours to be of average mass, that would mean a 
minimum total mass of the universe equal to 15,000,000,- 
000 terasuns or 100 gigagalaxies. 

Suppose, now, we work in the other direction. 

A milligram, or a thousandth of a gram, represents a 
quantity of matter easily visible to the naked eye. A drop 
of water would weigh about 50 milligrams. 

Drop to a microgram, or a millionth of a gram, and we 
are in the microscopic range. An amoeba would weigh in 
the neighborhood of five micrograms. 

The cells of our body are considerably smaller and for 
them we drop down to the nanogram, or a billionth of a 
gram. The average liver cell has a weight of about two 
nanograms. 

Below the cells are the viruses, but even if we drop to 
the picogram, a trillionth of a gram, we do not reach 
that realm. The tobacco-mosaic virus, for instance, weighs 
only 66 attograms. 

Nor is that particularly near the bottom of the scale. 
There are molecules far smaller than the smallest virus, 
and the atoms that make up the molecules and the par- 
ticles that make up the atom. Consider the following table: 

WEIGHT IN ATTOGRAMS 

hemoglobin molecule 0. 1 

uranium atom 0,0004 

proton 0.00000166 

electron 0,0000000009 


P re- Fixing It Up 


157 


All told, the range in mass from the electron to the 
minimum total mass of the known universe covers 83 
orders of magnitude. In other words, it would take 10 s3 
electrons to make a heap as massive as the total known 
universe. 

In some ways, time (the third of the types of measure- 
ment I am considering) possesses the most familiar units, 
because that is the one place where the metric system 
introduced no modification at all, We still have the second, 
the minute, the hour, the day, the year, and so on. 

This means, too, that the units of time are the only 
ones used by scientists that lack a systematic prefix system. 
The result is that you cannot tell, offhand, the number of 
seconds in a week or the number of minutes in a year or 
the number of days in fifteen years. Neither can scientists. 

The fundamental unit of time is the second and we 
could, if we wished, build the metric prefixes on those as 
follows: 


1 second 

equals 

1 second 

1 kitosecond 

equals 

16% minutes 

1 megasecond 

equals 

1 1% days 

1 gigasecond 

equals 

32 years 

1 terasecond 

equals 

32,000 years 


It is sobering to think that I have lived only a little 
over 134 gigaseconds *; that civilization has existed for at 
most about 250 gigaseconds; and that man-like creatures 
may not have existed for more than 18 teraseconds al- 
together. Still, that doesn’t make much of an inroad into 
geologic time and even less of an inroad into astronomic 
time. 

The solar system has been in existence for about 
150,000 teraseconds and may well remain in existence 
without major change for 500,000 additional teraseconds. 
The smaller the star, the more carefully it hoards its fuel 
supply and a red dwarf may last without undue change 
for as long as 3,000,000 teraseconds. As for the total age 
of the universe, past and future, I say nothing. There is 

[* Since this article first appeared, my age has increased to glgff' 
seconds f alas, but never mlnd t it's better than the alternative .] 



158 


NUMBERS AND MEASUREMENT 


AMOEBA 

The amoeba is a one-celled animal and in usually con- 
sidered the most primitive of the type . It has no fixed 
shape as other one-celled animals (“protozoa”) have but 
can bulge at any point to form a “ pseudopod ” ( Greek 
for “false foot”). It moves by means of these pseudopods 
and that is considered the most primitive form of animal 
locomotion. 

The fact that its shape is not fixed, but is changeable } 
is the basis of its name, which is from the Greek word 
for Kt change” The particular species of amoeba we com- 
monly mean when the name is used without qualification 
is “ Amoeba prole us” which is found on decaying organic 
matter in streams and ponds. The word “proteus” is the 
name of a Greek demigod who could change his shape 
at will. 

There are numerous other species of amoeba , some of 
which are parasitic, and six of which can parasitize man. 
One of them , Entamoeba histolytica (“ amoeba-within ; 
cell-dissolving ”), causes amoebic dysentery . 

Although the amoeba is mentioned in the article as the 
type of small organism , it is not (as also indicated) a 
small cell. The amoeba must , within its single cell T include 
all the machinery for the essential functions of life. A 
human cell , far more specialized , can afford to be smaller. 
Thus, an amoeba has 2,400 times the volume of a typical 
body cell and about 25,000 times the volume of the 
smallest human cell, the spermatozoon , 

The smallest free-living cells are the bacteria , and the 
amoeba has 210,000,000 times the volume of the smallest 
bacteria . 

The smallest objects that can be considered alive (al- 
though they function only within cells they parasitize) 
are the viruses. The amoeba has 2,400,000,000,000 the 
volume of the smallest virus. The amoeba is as large to 
that smallest virus as we are to the amoeba . 


Pre-Fixing It Up 


159 




160 


NUMBERS AND MEASUREMENT 


Pre-Fixing ft Up 


161 


no way of estimating, and the continuous-creation boys 
consider its lifetime to be eternal. * 

I have one suggestion to make for astronomic time, 
however (a suggestion which I don’t think is particularly 
original with me). The sun, according to reasonable esti- 
mates, revolves about the galactic center once every 
200,000,000 years. This we could call a “galactic year” 
or, better, a “gal-year.” (An ugly word, but never mindf) 
One galyear is equal to 6250 teraseconds. On the other 
hand, a “picogalyear” is equal to 1 hour and 45 minutes. 

If we stick to galyears then, the entire fossil record 
covers at most only 3 galyears; the total life of the solar 
system thus far is only 25 galyears; and the total life of 
a red dwarf as a red dwarf is perhaps 500 galyears. 

But now I’ve got to try the other direction, too, and 
see what happens for small units of time. Here at least 
there are no common units to confuse us. Scientists have 
therefore been able to use millisecond and microsecond 
freely, and now they can join to that nanosecond, pico- 
second, , femtosecond , and attosecond . 

These small units of time aren’t very useful in the 
macroscopic world. When a Gagarin or a Glenn circles 
the earth at 5 miles a second, he travels less than 9 
yards in a millisecond and less than a third of an inch in 
a microsecond. The earth itself, moving at a velocity of 
miles a second in its travels about the sun, moves 
only a little over an inch in a microsecond. 

In other words, at the microsecond level, ordinary 
motion is frozen out. However, the motion of light is 
more rapid than any ordinary motion, while the motion 
of some speeding subatomic particles is nearly as rapid 
as that of light. Therefore, let’s consider the small units 
of time in terms of light. 


DISTANCE COVERED BY LIGHT 


1 second 

186,200 

miles 

1 millisecond 

186 

miles 

1 microsecond 

327 

yards 

1 nanosecond 

1 

foot 

1 picosecond 

Mo inch 


[* Since this article written, the continuous creation theory has 
about been wiped out t and it isn’t Ukeiy that the Universe f in its present 
form , at least, is eternal .] 


Now, you may think that at picosecond levels subatomic 
motion and even light-propagation is “frozen.” After all, 
l dismissed earth’s motion as “frozen” when it moved an 
inch. How much more so, then, when thousandths of an 
inch are in question. 

However, there is a difference. The earth, in moving an 
inch, moves of its own diameter. A speeding 

subatomic particle moving at almost the speed of light for 
a distance Mo of an inch moves 120,000,000,000 times 
its own diameter. To travel a hundred and twenty billion 
times its own diameter, the earth would have to keep on 
going for 1,500,000 years. For Gagarin or Glenn to have 
traveled for a hundred and twenty billion times their own 
diameter, they would have had to stay in orbit a full year. 

A subatomic particle traveling Vm of an inch is therefore 
anything but “frozen," and has time to make a fabulous 
number of collisions with other subatomic particles or to 
undergo internal changes. As an example, neutral pions 
break down in a matter of 0.1 femtosecond after for- 
mation. 

What’s more, the omega-meson breaks down in some- 
thing like 0.0001 attosecond or, roughly, the time it 
would take light to cross the diameter of an atomic 
nucleus and back. 

The entire range of time, then, from the lifetime of an 
omega-meson to that of a red-dwarf star covers a range 
of 40 orders of magnitude. In other words, during the 
normal life of a red dwarf, some 10 10 omega-mesons 
have time to come into existence and break down, one 
after the other. 

To summarize, the measurable lengths cover a range 
of 41 order of magnitude, the measurable masses 83 orders 
of magnitude, and the measurable times 40 orders of 
magnitude. Clearly, we are not overdoing it in expanding 
the metric system from 6 to 30 orders of magnitude. 



Part IV 


NUMBERS AND 
THE CALENDAR 




THE 

DAYS OF 
OUR YEARS 


A group of us meet for an occasional evening of 
talk and nonsense, followed by coffee and doughnuts and 
one of the group scored a coup by persuading a well- 
known entertainer to attend the session. The well-known 
entertainer made one condition, however. He was not to 
entertain, or even be asked to entertain. This was agreed 
to.* 

Now there arose a problem. If the meeting were left to 
its own devices, someone was sure to begin badgering the 
entertainer. Consequently, other entertainment had to be 
supplied, so one of the boys turned to me and said, “Say, 
you know what? 1 ’ 

I knew what and I objected at once. I said, “How can 
I stand up there and talk with everyone staring at this 
other fellow in the audience and wishing he were up there 
instead? You'd be throwing me to the wolves!” 

But they all smiled very toothily and told me about 
the wonderful talks I give. (Somehow everyone quickly 
discovers the fact that I soften into putty as soon as the 
flattery is turned on.) In no time at all, I agreed to be 


[* / didn’t tiQme the entertainer when this article first appeared in Au- 
SMSt 1964 , because I thought he wouldn't want me to . / iwis wrong, 
because when 1 met him again months later ami asked for his autograph t 
he wrote "To Isaac, with best wishes, from a well-known entertainer 


165 



166 


NUMBERS AND THE CALENDAR 


The Days of Our Years 


167 


thrown to the wolves. Surprisingly, it worked, which 
speaks highly for the audience’s intellect — or perhaps 
their magnanimity. 

As it happened, the meeting was held on “leap day” 
and so my topic of conversation was ready-made and the 
gist of it went as follows: 

I suppose there’s no question but that the earliest unit 
of time-telling was the day. It forces itself upon the 
awareness of even the most primitive of humanoids. How- 
ever, the day is not convenient for long intervals of time. 
Even allowing a primitive lifespan of thirty years, a man 
would live some 11,000 days and it is very easy to lose 
track among all those days. 

Since the Sun governs the day-unit, it seems natural to 
turn to the next most prominent heavenly body, the Moon, 
for another unit. One offers itself at once, ready-made — 
the period of the phases. The Moon waxes from nothing 
to a full Moon and then to nothing in a definite period 
of time. This period of time is called the “month” in 
English (clearly from the word “moon”) or more spe- 
cifically, the “lunar month,” since we have other months, 
representing periods of time slightly shorter or slightly 
longer than the one that is strictly tied to the phases of 
the moon. 

The lunar month is roughly equal to 29V> days. More 
exactly, it is equal to 29 days, 12 hours, 44 minutes, 2.8 
seconds, or 29.5306 days. 

In pre-agricultural times, it may well have been that 
no special significance attached itself to the month, 
which remained only a convenient device for measuring 
moderately long periods of time. The life expectancy of 
primitive man was probably something like 350 months, 
which is a much more convenient figure than that of 
11,000 days. 

In fact, there has been speculation that the extended 
lifetimes of the patriarchs reported in the fifth chapter of 
the Book of Genesis may have arisen out of a confusion 
of years with lunar months. For instance, suppose Methu- 
selah had lived 969 lunar months. This would be just 
about 79 years, a very reasonable figure. However, once 
that got twisted to 969 years by later tradition we gained 
the “old as Methuselah” bit. 

However, I mention this only in passing, for this idea 


is not really taken seriously by any biblical scholars. It is 
much more likely that these lifetimes are a hangover 
from Babylonian tradition about the times before the 
Flood. . . - But I am off the subject. 

It is my feeling that the month gained a new and 
enhanced importance with the introduction of agriculture. 
An agricultural society was much more closely and pre- 
cariously tied to the seasons than a hunting or herding 
society was. Nomads could wander in search of grain or 
grass but farmers had to stay where they were and hope 
for rain. To increase their chances, farmers had to be cer- 
tain to sow at a proper time to take advantage of seasonal 
rains and seasonal warmth; and a mistake in the sowing 
period might easily spell disaster. What’s more, the de- 
velopment of agriculture made possible a denser popula- 
tion, and that intensified the scope of the possible disaster. 

Man had to pay attenion, then, to the cycle of seasons, 
and while he was still in the prehistoric stage he must 
have noted that those seasons came full cycle in roughly 
twelve months. In other words, if crops were planted at a 
particular time of the year and all went well, then if 
twelve months were counted from the first planting and 
crops were planted again, all would again go well. 

Counting the months can be tricky in a primitive so- 
ciety, especially when a miscount can be ruinous, so it 
isn’t surprising that the count was usually left in the hands 
of a specialized caste, the priesthood. The priests could not 
only devote their time to accurate counting, but could 
also use their experience and skill to propitiate the gods. 
After all, the cycle of the seasons was by no means as 
rigid and unvarying as was the cycle of day and night or 
the cycle of the phases of the moon. A late frost or a 
failure of rain could blast that season’s crops, and since 
such flaws in weather were bound to follow any little 
mistake in ritual (at least so men often believed), the 
priestly functions were of importance indeed. 

It is not surprising then, that the lunar month grew to 
have enormous religious significance. There were new 
Moon festivals and special priestly proclamations of each 
one of them, so that the lunar month came to be called 
the “synodic month.” 

The cycle of season* is called the “year” and twelve 
lunar months therefore make up a “lunar year.” The use 



168 


NUMBERS AND THE CALENDAR 


THE CRESCENT MOON 

The crescent Moon, which marked the beginning of the 
month in ancient times , together with the remaining 
phases of the Moon } was responsible for the birth of 
astronomy T for surely the regularly changing shape of the 
Moon war the first object in the sky that roused man's 
curiosity . The necessities and value of calendar-making 
must have urged man on to develop mathematics and 
religion out of the lunar cycle. 

There was something else , too . - . . 

The ancient Greek philosophers found it aesthetically 
satisfying to divide the Universe into two parts: the Earth 
and the heavenly bodies. To do so, they sought for funda- 
mental differences in properties. Thus: The heavenly 
bodies were all luminous , while the earth was nonlumin- 
ous. 

The Moon , however f had to be an exception to this 
general rule. The relationship of the phases of the Moon 
to the relative positions of the Moon and Sun made it 
clear even in ancient times that the Moon shone only by 
reflected sunlight. That meant that , of its own, the Moon 
was as dull and nonluminous as the Earth . 

What's more , when the Moon is in its crescent phase 
and is just a thin sliver of curling light , as in the illustration f 
the rest of the Moon is sometimes seen shining with a 
dim ruddy light of its own „ Galileo pointed out that from 
the Moon , the Earth was seen in the full phase and that 
the Moon was shining dimly in Earthlight. Earth, too , 
reflected light and war as luminous as the Moon. 

Then , too, the ancient Greeks had already determined 
the distance of the Moon quite accurately, and it could 
be seen to be a world of some two thousand miles in 
diameter to appear to be as large as it seemed from that 
distance. In short, thanks to the Moon, naked-eye astron- 
omy sufficed to establish the doctrine of Tt plurality of 
worlds," for if the Moon war a world so might many 
other heavenly bodies be. 



170 


NUMBERS AND THE CALENDAR 


The Days of Our Years 


171 


of lunar years in measuring time is referred to as the use 
of a 'lunar calendar. " The only important group of peo- 
ple in modern times, using a strict lunar calendar, are the 
Mohammedans. Each of the Mohammedan years is made 
up of 12 months which are, in turn, usually made up of 
29 and 30 days in alternation. 

Such months average 29.5 days, but the length of the 
true lunar month is, as I've pointed out, 29.5306 days. 
The lunar year built up out of twelve 29. 5 -day months is 
354 days long, whereas twelve lunar months are actually 
354.37 days long. 

You may say “So what" but don't. A true lunar year 
should always start on the day of the new Moon. If, how- 
ever, you start one lunar year on the new Moon and then 
simply alternate 2 9 -day and 30-day months, the third 
year will start the day before the new Moon, and the 
sixth year will start two days before the new Moon. To 
properly religious people, this would be unthinkable. 

Now it so happens that 30 true lunar years come out 
to be almost exactly an even number of days — 10,631.016. 
Thirty years built up out of 29. 5 -day months come to 
10,620 days — just 11 days short of keeping time with the 
Moon. For that reason, the Mohammedans scatter 11 
days through the 30 years in some fixed pattern which 
prevents any individual years from starting as much as a 
full day ahead or behind the New Moon. In each 30-year 
cycle there are nineteen 354-day years and eleven 355-day 
years, and the calendar remains even with the Moon. 

An extra day, inserted in this way to keep the calendar 
even with the movements of a heavenly body, is called 
an "intercalary day"; a day inserted "between the calendar," 
so to speak. 

The lunar year, whether it is 354 or 355 days in length, 
does not, however, match the cycle of the seasons. By 
the dawn of historic times the Babylonian astronomers 
had noted that the Sun moved against the background 
of stars. This passage was followed with absorption be- 
cause it grew apparent that a complete circle of the sky 
by the Sun matched the complete cycle of the seasons 
closely. (This apparent influence of the stars on the 
seasons probably started the Babylonian fad of astrology 
— which is still with us today. ) 

The Sun makes its complete cycle about the zodiac in 
roughly 365 days, so that the lunar year is about 1 1 days 


shorter than the season-cycle, or “solar-year." Three lunar 
years fall 33 days, or a little more than a full month, 
behind the season-cycle. 

This is important. If you use a lunar calendar and start 
it so that the first day of the year is planting time, then 
three years later you are planting a month too soon, and 
by the time a decade has passed you are planting in 
midwinter. After 33 years the first day of the year is back 
where it is supposed to be, having traveled through the 
entire solar year. 

This is exactly what happens in the Mohammedan year. 
The ninth month of the Mohammedan year is named 
Ramadan, and it is especially holy because it was the 
month in which Mohammed began to receive the revela- 
tion of the Koran, In Ramadan, therefore, Moslems abstain 
from food and water during the day-light hours. But each 
year, Ramadan falls a bit earlier in the cycle of the 
seasons, and at 3-year intervals it is to be found in the 
hot season of the year; at this time abstaining from drink 
is particularly wearing, and Moslem tempers grow par- 
ticularly short. 

The Mohammedan years are numbered from the Hegira; 
that is, from the date when Mohammed fled from Mecca 
to Medina, That event took place in a.d. 622. Ordinarily, 
you might suppose, therefore, that to find the number of 
the Mohammendan year, one need only subtract 622 
from the number of the Christian year. This is not quite 
so, since the Mohammedan year is shorter than ours. I 
write this chapter in a.d. 1964 and it is now 1342 solar 
years since the Hegira. However, it is 1384 lunar years 
since the Hegira, so that, as I write, the Moslem year 
is a.h, 1384. 

I’ve calculated that the Mohammedan year will catch 
up to the Christian year in about nineteen millennia. The 
year a.d. 20,874 will also be a.h. 20,874, and the Moslems 
will then be able to switch to our year with a minimum 
of trouble. 

But what can we do about the lunar year in order to 
make it keep even with the seasons and the solar year? 
We can't just add 11 days at the end, for then the next 
year would not start with the new Moon and to the 
ancient Babylonians, for instance, a new Moon start was 
essential 



172 


NUMBERS AND THE CALENDAR 


However, if we start a solar year with the new Moon 
and wait, we will find that the twentieth solar year there- 
after starts once again on the day of the new Moon, You 
see, 19 solar years contain just about 235 lunar months. 

Concentrate on those 235 lunar months. That is equiva- 
lent to 19 lunar years (made up of 12 lunar months each) 
plus 7 lunar months left over. We could, then, if we 
wanted to, let the lunar years progress as the Moham- 
medans do, until 19 such years had passed. At this time 
the calendar would be exactly 7 months behind the seasons, 
and by adding 7 months to the 19th year (a 19th year 
of 19 months— very neat) we could start a new 19-year 
cycle, exactly even with both the Moon and the seasons. 

The Babylonians were unwilling, however, to let them- 
selves fall 7 months behind the season. Instead, they added 
that 7-month discrepancy through the 19-year cycle, one 
month at a time and as nearly evenly as possible. Each 
cycle had twelve 12-month years and seven 13 -month 
years. The “intercalary month” was added in the 3rd, 
6th, 8th, 11th, 14th, 17th, and 19th year of each cycle, 
so that the year was never more than about 20 days be- 
hind or ahead of the Sun. 

Such a calendar, based on the lunar months, but gim- 
micked so as to keep up with the Sun, is a “lunar-solar 
calendar.” 

The Babylonian lunar-solar calendar was popular in 
ancient times since it adjusted the seasons while preserving 
the sanctity of the Moon. The Hebrews and Greeks both 
adopted this calendar and, in fact, it is still the basis for 
the Jewish calendar today. The individual dates in the 
Jewish calendar are allowed to fall slightly behind the Sun 
until the intercalary month is added, when they suddenly 
shoot slightly ahead of the Sun. That is why holidays like 
Passover and Yom Kippur occur on different days of 
the civil calendar (kept strictly even with the Sun) each 
year. These holidays occur on the same day of the year 
each year in the Jewish calendar. 

The early Christians continued to use the Jewish calen- 
dar for three centuries, and established the day of Easter 
on that basis. As the centuries passed, matters grew some- 
what complicated, for the Romans (who were becoming 
Christian in swelling numbers) were no longer used to a 
lunar-solar calendar and were puzzled at the erratic jump- 
ing about of Easter. Some formula had to be found by 


The Days of Our Years 173 

which the correct date for Easter could be calculated in 
advance, using the Roman calendar. 

It was decided at the Council of Nicaea, in a.d. 325 
(by which time Rome had become officially Christian), 
that Easter was to fall on the Sunday after the first full 
Moon after the vernal equinox, the date of the vernal 
equinox being established as March 21. However, the full 
Moon referred to is not the actual full Moon, but a ficti- 
tious one called the “Paschal Full Moon” (“Paschal” 
being derived from Pesach , which is the Hebrew word for 
Passover). The date of the Paschal Full Moon is calculated 
according to a formula involving Golden Numbers and 
Dominical Letters, which I won't go into. 

The result is that Easter still jumps about the days of 
the civil year and can fall as early as March 22 and as 
late as April 25. Many other church holidays are tied to 
Easter and likewise move about from year to year. 

Moreover, all Christians have not always agreed on the 
exact formula by which the date of Easter was to be 
calculated. Disagreement on this detail was one of the 
reasons for the schism between the Catholic Church of 
the West and the Orthodox Church of the East. In the 
early Middle Ages there was a strong Celtic Church which 
had its own formula. 

Our own calendar is inherited from Egypt, where sea- 
sons were unimportant. The one great event of the year 
was the Nile flood, and this took place (on the average) 
every 365 days. From a very early date, certainly as early 
as 2781 b.c., the Moon was abandoned and a “solar 
calendar,” adapted to a constant-length 365-day year, 
was adopted. 

The solar calendar kept to the tradition of 12 months, 
however. As the year was of constant length, the months 
were of constant length, too — 30 days each. This meant 
that the new Moon could fall on any day of the month, 
but the Egyptians didn’t care. (A month not based on the 
Moon is a “calendar month.”) 

Of course 12 months of 30 days each add up only to 
360 days, so at the end of each 12-month cycle, 5 addi- 
tional days were added and treated as holidays. 

The solar year, however, is not exactly 365 days long. 
There are several kinds of solar years, differing slightly in 



174 


NUMBERS AND THE CALENDAR 


The Days of Our Years 


175 


length, but the one upon which the seasons depend is the 
“tropical year,” and this is about 3 65H days long. 

This means that each year, the Egyptian 365-day year 
falls 14 day behind the Sun. As time went on the Nile 
flood occurred later and later in the year, until finally it 
had made a complete circuit of the year. In 1460 tropica] 
years, in other words, there would be 1461 Egyptian years. 

This period of 1461 Egyptian years was called the 
“Sothic cycle,” from Sothis, the Egyptian name for the 
star Sirius. If, at the beginning of one Sothic cycle, Sirius 
rose with the Sun on the first day of the Egyptian year, 
it would rise later and later during each succeeding year 
until finally, 1461 Egyptian years later, a new cycle would 
begin as Sothis rose with the Sun on New Year's Day 
once more. 

The Greeks had learned about that extra quarter day 
as early as 380 B.c., when Eudoxus of Cnidus made the 
discovery. In 239 b.c, Ptolemy Euergetes, the Macedonian 
king of Egypt, tried to adjust the Egyptian calendar to 
take that quarter day into account, but the ultra-conserva- 
tive Egyptians would have none of such a radical inno- 
vation. 

Meanwhile, the Roman Republic had a lunar-solar calen- 
dar, one in which an intercalary month was added every 
once in a while. The priestly officials in charge were 
elected politicians, however, and were by no means as 
conscientious as those in the East. The Roman priests 
added a month or not according to whether they wanted 
a long year (when the other annually elected officials in 
power were of their own party) or a short one (when 
they were not). By 46 b.c., the Roman calendar was 80 
days behind the Sun. 

Julius Caesar was in power then and decided to put an 
end to this nonsense. He had just returned from Egypt 
where he had observed the convenience and simplicity 
of a solar year, and he imported an Egyptian astronomer, 
Sosigenes, to help him. Together, they let 46 b.c. continue 
for 445 days so that it was later knowm as “The Year of 
Confusion,” However, this brought the calendar even 
with the Sun so that 46 b.c. was the last year of confu- 
sion. 

With 45 b.c. the Romans adopted a modified Egyptian 
calendar in which the five extra days at the end of the 


year were distributed throughout the year, giving us our 
months of uneven length. Ideally, we should have seven 
30-day months and five 31-day months. Unfortunately 
the Romans considered February an unlucky month and 
shortened it, so that we ended with a silly arrangement of 
seven 31 -day months, four 30-day months, and one 28-day 
month. 

In order to take care of that extra % day, Caesar and 
Sosigenes established every fourth year with a length of 
366 days, (Under the numbering of the years of the 
Christian era, every year divisible by 4 has the intercalary 
day — set as February 29. Since 1964 divided by 4 is 491, 
without a remainder, there is a February 29 in 1964.) 

This is the “Julian year,” after Julius Caesar. At the 
Council of Nicaea, the Christian Church adopted the 
Julian calendar. Christmas was finally accepted as a 
Church holiday after the Council of Nicaea and was 
therefore given a date in the Julian year. It does not, 
therefore, bounce about from year to year as Easter does. 

The 365-day year is just 52 weeks and 1 day long. This 
means that if February 6, for instance, is on a Sunday in 
one year, it is on a Monday the next year, on a Tuesday 
the year after, and so on. If there were only 365-day 
years, then any given date would move through the days 
of the week in steady progression. If a 366-day year is 
involved, however, that year is 52 weeks and 2 days long, 
and if February 6 is on Tuesday that year, it is on 
Thursday the year after. The day has leaped over Wed- 
nesday. It is for that reason that the 366-day year is called 
“leap year” and February 29 is “leap day,” 

All would have been well if the tropical year were 
really exactly 365.25 days long; but it isn't. The tropical 
year is 365 days, 5 hours, 48 minutes, 46 seconds, or 
365.24220 days long. The Julian year is, on the average, 
11 minutes 14 seconds, or 0.0078 day, too long. 

This may not seem much, but it means that the Julian 
year gains a full day on the tropical year in 128 years. 
As the Julian year gains, the vernal equinox, falling be- 
hind, comes earlier and earlier in the year. At the Coun- 
cil of Nicaea in a.d. 325, the vernal equinox was on 
March 21, By a.d. 453 it was on March 20, by a.d. 581 
on March 19, and so on. By a.d. 1263, in the lifetime of 



176 


NUMBERS AND THE CALENDAR 


JULIUS CAESAR 


Julius Caesar , for whom the Julian calendar is named , is, 
of course, far better known among the general public for 
many other reasons . 

He was born in 102 b.c,, and he was just about the 
most remarkable man of ancient times . He urn a man 
of enormous courage, a playboy and wastrel , who, in 
middle life, turned to leading armies and proved himself 
to be a great general who never lost a battle. He was a 
great orator, second only to Cicero among the Romans t 
and a great writer. And he was a successful politician. 

His charm vim legendary. In 76 B.c. he set sail for 
the island of Rhodes in order to study under the best 
Greek teachers . On the way , he was captured by pirates 
who held him for ransom of about $100,000 in modern 
money . While the money was being scraped up by friends 
and relatives, Caesar charmed his captors and had a 
great time with them . While they were engaged in friendly 
conversation , Caesar told them that once he was set free , 
he would return with a fleet and hang every one of them. 
The pirates laughed at the joke, and when Caesar was 
paid for and freed r he did indeed return with a fleet and 
hang them all 

With the Roman Republic slowly decaying as it proved 
increasingly difficult to rule the empire it had gathered , 
Caesar engaged in civil war (in the course of which he 
entered Egypt and had a famous love affair with Cleo- 
patra) and finally emerged as sole ruler and dictator of 
the Roman realm. 

Here was where his own great failing showed up. He 
firmly believed an enemy forgiven was an enemy destroyed. 
He forgave many vjHo had fought on the other side and 
gave them high positions in the state . They conspired 
against him , and on March 75, 44 b.c. ( the Ides of 
March), they assassinated him . 


178 


NUMBERS AND THE CALENDAR 


The Days of Our Years 


179 


Roger Bacon, the Julian year had gained eight days on 
the Sun and the vernal equinox was on March 13. 

Still not fatal, but the Church looked forward to an 
indefinite future and Easter was tied to a vernal equinox 
at March 21. If this were allowed to go on, Easter would 
come to be celebrated in midsummer, while Christmas 
would edge into the spring. In 1263, therefore, Roger 
Bacon wrote a letter to Pope Urban IV explaining the 
situation. The Church, however, took over three centuries 
to consider the matter. 

By 1582 the Julian calendar had gained two more days 
and the vernal equinox was falling on March 11. Pope 
Gregory XIII finally took action. First, he dropped ten 
days, changing October 5, 1852 to October 15, 1582. 
That brought the calendar even with the Sun and the 
vernal equinox in 1583 fell on March 21 as the Council 
of Nicaea had decided it should. 

The next step was to prevent the calendar from getting 
out of step again. Since the Julian year gains a full day 
every 128 years, it gains three full days in 384 years or, 
to approximate slightly, three full days in four centuries. 
That means that every 400 days, three leap years (accord- 
ing to the Julian system) ought to be omitted. 

Consider the century years — 1500, 1600, 1700, and so 
on. In the Julian year, all century years are divisible by 4 
and are therefore leap years. Every 400 years there are 
4 such century years, so why not keep 3 of them ordinary 
years, and allow only one of them (the one that is divisible 
by 400) to be a leap year? This arrangement will match 
the year more closely to the Sun and give us the “Gre- 
gorian calendar.” 

To summarize: Every 400 years, the Julian calendar 
allows 100 leap years for a total of 146,100 days. In that 
same 400 years, the Gregorian calendar allows only 97 
leap years for a total of 146,097 days. Compare these 
lengths with that of 400 tropical years, which comes to 
146,096.88. Whereas, in that stretch of time, the Julian 
year had gained 3.12 days on the Sun, the Gregorian 
year had gained only 0.12 day. 

Still, 0.12 day is nearly 3 hours, and this means that 
in 3400 years the Gregorian calendar will have gained a 
full day on the Sun. Around a.d, 5000 we will have to 
consider dropping out one extra leap year. 


But the Church had waited a little too long to take 
action. Had it done the job a century earlier, all western 
Europe would have changed calendars without trouble. 
By a.d. 1582, howe\er, much of northern Europe had 
turned Protestant. These nations would far sooner remain 
out of step with the Sun in accordance with the dictates 
of the pagan Caesar, than consent to be corrected by the 
pope. Therefore they kept the Julian year. 

The year 1600 introduced no crisis. It was a century 
year but one that was divisible by 400. Therefore, it was 
a leap year by both the Julian and Gregorian calendars. 
But 1700 was a different matter. The Julian calendar had 
it as a leap year and the Gregorian did not. By March 1, 
1700, the Julian calendar was going to be an additional 
day ahead of the Sun (eleven days altogether). Denmark, 
the Netherlands, and Protestant Germany gave in and 
adopted the Gregorian calendar. 

Great Britain and the American colonies held out until 
1752 before giving in. Because of the additional day 
gained in 1700, they had to drop eleven days and changed 
September 2, 1752 to September 13, 1752. There were 
riots all over England as a result, for many people came 
quickly to the conclusion that they had suddenly been 
made eleven days older by legislation. 

“Give us back our eleven days!” they cried in despair. 

(A more rational objection was the fact that although 
the third quartetr of 1752 was short eleven days, land- 
lords calmly charged a full quarter’s rent.) 

As a result of this, it turns out that Washington was 
not bom on “Washington’s birthday.” He was born on 
February 22, 1732 on the Gregorian calendar, to be sure, 
but the date recorded in the family Bible had to be the 
Julian date, February 11, 1732. When the changeover 
took place, Washington— a remarkably sensible man — 
changed the date of his birthday and thus preserved the 
actual day. 

The Eastern Orthodox nations of Europe were more 
stubborn than the Protestant nations. The years 1800 and 
1900 went by. Both were leap years by the Julian calendar, 
but not by the Gregorian calendar. By 1900, then, the 
Julian vernal equinox was on March 8 and the Julian 
calendar was 13 days ahead of the Sun. It was not until 
after World War I that the Soviet Union, for instance, 
adopted the Gregorian calendar. (In doing so, the Soviets 



180 


NUMBERS AND THE CALENDAR 


made a slight modification of the leap year pattern which 
made matters even more accurate. The Soviet calendar 
will not gain a day on the Sun until fully 35,000 years 
pass. ) 

Some of the Orthodox churches, however, still cling to 
the Julian year, which is why the Orthodox Christmas 
falls on January 6 on our calendar* It is still December 
25 by their calendar. 

In fact, a horrible thought occurs to me — 

I was myself born at a time when the Julian calendar 
was still in force in the— ahem— old country.* Unlike 
George Washington, I never changed the birthdate and, 
as a result, each year I celebrate my birthday 13 days 
earlier than I should, making myself 13 days older than 
I have to be. 

And this 13-day older me is in all the records and I 
can’t ever change it back. 

Give me back my 13 days! Give me back my 13 days! 
Give me back , . . 


* Welt, the Soviet Union , if you must know . I came here at the age of 5 . 


BEGIN 
AT THE 
BEGINNING 



Each year, another New Year's Day falls upon us; 
and because my birthday follows hard upon New Year’s 
Day, the beginning of the year is always a doubled occa- 
sion for great and somber soul-searching on my part. 

Perhaps I can make my consciousness of passing time 
less poignant by thinking more objectively. For instance, 
who says the year starts on New Year’s Day? What is there 
about New Year’s Day that is different from any other 
day? What makes January 1 so special? 

In fact, when we chop up time into any kind of units, 
how do we decide with which unit to start? 

For instance, let’s begin at the beginning fas I dearly 
love to do) and consider the day itself. 

The day is composed of tw r o parts, the daytime * and 
the night. Each, separately, has a natural astronomic be- 
ginning. The daytime begins with sunrise; the night begins 
with sunset. (Dawn and twilight encroach upon the night 
but that is a mere detail ) 

* It is very annoying that "day” means both the sunlit portion of time 
and the twenty-four-hour period of daytime and night together* This is a 
completely unnecessary shortcoming of the admirable English language. 
I understand that the Greek language contains separate words for the two 
entities. I shall use " daytime " for the sunlit period and "day” for the 
twenty-four-hour period. 


181 



182 


NUMBERS AND THE CALENDAR 


Begin at the Beginning 


183 


In the latitudes in which most of humanity live* how- 
ever, both daytime and night change in length during the 
year (one growing longer as the other grows shorter) and 
there is* therefore, a certain convenience in using daytime 
plus night as a single twenty-four-hour unit of time. The 
combination of the two, the day, is of nearly constant 
duration. 

Well, then, should the day start at sunrise or at sunset? 
You might argue for the first, since in a primitive society 
that is when the workday begins. On the other hand, in 
that same society sunset is when the workday ends, and 
surely an ending means a new beginning. 

Some groups made one decision and some the other. 
The Egyptians, for instance, began the day at sunrise, 
while the Hebrews began it at sunset. 

The latter state of affairs is reflected in the very first 
chapter of Genesis in which the days of creation are de- 
scribed. In Genesis 1:5 it is written: “And the evening and 
the morning were the first day.” Evening (that is, night) 
comes ahead of morning (that is, daytime) because the 
day starts at sunset. 

This arrangement is maintained in Judaism to this day, 
and Jewish holidays still begin “the evening before." 
Christianity began as an offshoot of Judaism and remnants 
of this sunset beginning cling even now to some non- 
Jewish holidays. 

The expression Chritmas Eve, if taken literally, is the 
evening of December 25, but as we ail know it really 
means the evening of December 24 — which it would 
naturally mean if Christmas began “the evening before" 
as a Jewish holiday would. The same goes for New Year’s 
Eve. 

Another familiar example is All Hallows’ Eve, the eve- 
ning of the day before All Hallows’ Day, which is given 
over to the commemoration of all the “hallows” (or 
“saints”). All Hallows’ Day is on November 1, and All 
Hallows’ Eve is therefore on the evening of October 31. 
Need I tell you that All Hallows’ Eve is better known by 
its familiar contracted form of “Halloween.” 

As a matter of fact, though, neither sunset nor sunrise 
is now the beginning of the day. The period from sunrise 
to sunrise is slightly more than 24 hours for half the 
year as the daytime periods grow shorter, and slightly 
less than 24 hours for the remaining half of the year as 


the daytime periods grow longer. This is also true for the 
period from sunset to sunset. 

Sunrise and sunset change in opposite directions, either 
approaching each other or receding from each other, so 
that the middle of daytime (midday) and the middle of 
night (midnight) remain fixed at 24-hour intervals 
throughout the year. (Actually, there are minor deviations 
but these can be ignored.) 

One can begin the day at midday and count on a 
steady 24-hour cycle, but then the working period is split 
between two different dates. Far better to start the day at 
midnight when all decent people are asleep; and that, in 
fact, is what we do. 

Astronomers, who are among the indecent minority not 
in bed asleep at midnight, long insisted on starting their 
day at midday so as not to break up a night’s observation 
into two separate dates. However, the spirit of conformity 
was not to be withstood, and in 1925, they accepted the 
inconvenience of a beginning at midnight in order to get 
into step with the rest of the world. 

All the units of time that are shorter than a day depend 
on the day and offer no problem. You start counting the 
hours from the beginning of the day; you start counting 
the minutes from the beginning of the hours, and so on. 

Of course, when the start of the day changed its posi- 
tion, that affected the counting of the hours. Originally, 
the daytime and the night were each divided into twelve 
hours, beginning at, respectively, sunrise and sunset. The 
hours changed length with the change in length of daytime 
and night so that in June (in the northern hemisphere) the 
daytime was made up of twelve long hours and the night 
of twelve short hours, while in December the situation was 
reversed. 

This manner of counting the hours still survives in the 
Catholic Church as “canonical hours.” Thus, “prime" 
(“one”) is the term for 6 a.m. “Tierce” (“three") is 9 
a.m., “sext” (“six”) is 12 a.m., and “none" (“nine”) is 
3 p.m. Notice that “none" is located in the middle of the 
afternoon when the day is warmest. The warmest part of 
the day might well be felt to be the middle of the day, 
and the word was somehow switched to the astronomic 
midday so that we call 12 a.m. “noon.” 



184 


NUMBERS AND THE CALENDAR 


Begin at the Beginning 


185 


This older method of counting the hours also plays a 
part in one of the parables of Jesus (Matt. 20:1-16), in 
which laborers are hired at various times of the day, up 
to and including “the eleventh hour.” The eleventh hour 
referred to in the parable is one hour before sunset when 
the working day ends. For that reason, “the eleventh 
hour” has come to mean the last moment in which some- 
thing can be done. The force of the expression is lost on 
us, however, for we think of the eleventh hour as being 
either 11 a.m. or 11 p.m., and 11 a.m. is too early in the 
day to begin a feel panicky, while 1 1 p.m. is too late — 
we ought to be asleep by then. 

The week originated in the Babylonian calendar where 
one day out of seven was devoted to rest. (The rationale 
was that it was an unlucky day.) 

The Jews, captive in Babylon in the sixth century b.c., 
picked up the notion and established it on a religious 
basis, making it a day of happiness rather than of ill for- 
tune. They explained its beginnings in Genesis 2:2 where, 
after the work of the six days of creation — “on the seventh 
day God ended his work which he had made; and he 
rested on the seventh day.” 

To those societies which accept the Bible as a book of 
special significance, the Jewish “sabbath” (from the He- 
brew word for “rest”) is thus defined as the seventh, and 
last, day of the week. This day is the one marked Satur- 
day on our calendars, and Sunday, therefore, is the first 
day of a new week. All our calendars arrange the days in 
seven columns with Sunday first and Saturday seventh. 

The early Christians began to attach special significance 
to the first day of the week. For one thing, it was the 
“lord's day” since the Resurrection had taken place on a 
Sunday. Then, too, as time went on and Christians began 
to think of themselves as something more than a Jewish 
sect, it became important to them to have distinct rituals 
of their own. In Christian societies, therefore, Sunday, 
and not Saturday, became the day of rest, (Of course, in 
our modem effete times, Saturday and Sunday are both 
days of rest, and are lumped together as the “weekend,” 
a period celebrated by automobile accidents.) 

The fact that the work week begins on Monday causes 
a great many people to think of that as the first day of 
the week, and leads to the following children’s puzzle 


(which I mention only because it trapped me neatly the 
first time X heard it) . 

You ask your victim to pronounce t-o, t-o-o, and t-w-o, 
one at a time, thinking deeply between questions. In each 
case he says (wondering what’s up) “tooooo.” 

Then you say, “Now pronounce the second day of the 
week” and his face clears up, for he thinks he sees the 
trap. He is sure you are hoping he will say “toooosday” 
like a lowbrow. With exaggerated precision, therefore, he 
says “tyoosday.” 

At which you look gently puzzled and say, “Isn’t that 
strange? I always pronounce it Monday.” 

The month, being tied to the Moon, began, in ancient 
times, at a fixed phase. In theory, any phase will do. The 
month can start at each full Moon, or each first quarter, 
and so on. Actually, the most logical way is to begin 
each month with the new Moon— that is, on that evening 
when the first sliver of the growing crescent makes itself 
visible immediately after sunset. To any logical primitive, 
a new Moon is clearly being created at that time and the 
month should start then. 

Nowadays, however, the month is freed of the Moon 
and is tied to the year, which is in turn based on the Sun. 
In our calendar, in ordinary years, the first month begins 
on the first day of the year, the second month on the 
32nd day of the year, the third month on the 60th day 
of the year, the fourth month on the 91st day of the year, 
and so on — quite regardless of the phases of the Moon. 
(In a leap year, all the months from the third onward 
start a day late because of the existence of February 29. ) 

But that brings us to the year. When does that begin 
and why? 

Primitive agricultural societies must have been first 
aware of the year as a succession of seasons. Spring, sum- 
mer, autumn, and winter were the morning, midday, eve- 
ning, and night of the year, as in the case of the day, 
there seemed two equally qualified candidates for the post 
of beginning. 

The beginning of the work year is the time of spring, 
when warmth returns to the earth and planting can begin. 
Should that not also be the beginning of the year in 
general? On the other hand, autumn marks the end of the 



186 


NUMBERS AND THE CALENDAR 


Begin at the Beginning 


187 


work year, with the harvest (it is to be devoutly hoped) 
safely in hand. With the work year ended, ought not the 
new year begin? 

With the development of astronomy, the beginning of 
the spring season was associated with the vernal equinox 
which, on our calendar, falls on March 20, while the 
beginning of autumn is associated with the autumnal 
equinox which falls, half a year later, on September 23. 

Some societies chose one equinox as the beginning 
and some the other. Among the Hebrews, both equinoxes 
came to be associated with a New Year’s Day. One of 
these fell on the first day of the month of Nisan (which 
comes at about the vernal equinox) . In the middle of that 
month comes the feast of Passover, which is thus tied to 
the vernal equinox. 

Since, according to the Gospels, Jesus’ Crucifixion and 
Resurrection occurred during the Passover season (the 
Last Supper was a Passover seder), Good Friday and 
Easter are also tied to the vernal equinox (see Chapter 11). 

The Hebrews also celebrated a New Year’s Day on the 
first two days of Tishri (which falls at about the autumnal 
equinox), and this became the more important of the two 
occasions. It is celebrated by Jews today as "Rosh Ha- 
shonah” ("head of the year”), the familiarly known 
"Jewish New Year.” 

A much later example of a New Year’s Day in con- 
nection with the autumnal equinox came in connection 
with the French Revolution. On September 22, 1792, the 
French monarchy was abolished and a republic pro- 
claimed. The Revolutionary idealists felt that since a new 
epoch in human history had begun, a new calendar was 
needed. They made September 22 the New Year’s Day and 
established a new list of months. The first month was 
Vendemiare, so that September 22 became Vendemiare 1. 

For thirteen years, Vendemiare 1 continued to be the 
official New Year’s Day of the French Government, but 
the calendar never caught on outside France or even 
among the people inside France. In 1806 Napoleon gave 
up the struggle and officially reinstated the old calendar. 

There are two important solar events in addition to the 
equinoxes. After the vernal equinox, the noonday Sun 
continues to rise higher and higher until it reaches a 
maximum height on June 21, which is the summer solstice, 


and this day, in consequence, has the longest daytime 
period of the year. 

The height of the noonday Sun declines thereafter 
until it reaches the position of the autumnal equinox. 
It then continues to decline farther and farther till it 
reaches a minimum height on December 21, the winter 
solstice and the shortest daytime period of the year. 

The summer solstice is not of much significance. "Mid- 
summer Day” falls at about the summer solstice (the 
traditional English day is June 24). This is a time for 
gaiety and carefree joy, even folly. Shakespeare’s A Mid- 
summer Night’s Dream is an example of a play devoted 
to the kind of not-to-be-taken-seriously fun of the season, 
and the phrase "midsummer madness” may have arisen 
similarly. 

The winter solstice is a much more serious affair. The 
Sun is declining from day to day, and to a primitive 
society, not sure of the invariability of astronomical laws, 
it might welt appear that this time, the Sun will continue 
its decline and disappear forever so that spring will never 
come again and all life will die. 

Therefore, as the Sun’s decline slowed from day to 
day and came to a halt and began to turn on December 
21, there must have been great relief and joy which, in 
the end, became ritualized into a great religious festival, 
marked by gaiety and licentiousness. 

The best-known examples of this are the several days 
of holiday among the Romans at this season of the year. 
The holiday was in honor of Saturn (an ancient Italian 
god of agriculture) and was therefore called the "Sat- 
urnalia.” It was a time of feasting and of giving of 
presents; of good will to men, even to the point where 
slaves were given temporary freedom while their masters 
waited upon them. There was also a lot of drinking at 
Saturnalia parties. 

In fact, the word “saturnalian” has come to mean dis- 
solute, or characterized by unrestrained merriment. 

There is logic, then, in beginning the year at the winter 
solstice which marks, so to speak, the birth of a new Sun, 
as the first appearance of a crescent after sunset marks 
the birth of a new Moon. Something like this may have 
been in Julius Caesar's mind when he reorganized the 
Roman calendar and made it solar rather than lunar (see 
Chapter 11). 



188 


NUMBERS AND THE CALENDAR 


NAPOLEON BONAPARTE 

Napoleon , who went from Corsican rebel , to French gen- 
eral, to Emperor , to exile r is mentioned briefly in this 
article as having put an end to the only modern experi- 
ment in novel calendars. He was distantly involved in 
science in other respects . 

In 1807 1 when his conquests brought him to Poland , 
he expressed surprise that no statue to Copernicus had 
ever been erected , and one was put up in consequence. 
When it was, no Catholic priest would agree to officiate 
on the occasion. 

Napoleon patronized scientists such as Lagrange and 
Laplace , promoted them and honored them. Once , when 
he was holding British prisoners of war s he released them 
only after Edward Jenner (the discoverer of vaccination 
against smallpox) added his name to those petitioning for 
the release. 

When Napoleon invaded Egypt in 1798 > he brought a 
number of scientists with him to investigate its ancient 
civilization. The Rosetta stone f inscribed in both Greek 
and Egyptian, was discovered on that occasion, and 
Egyptian was eventually deciphered so that our knowl- 
edge of ancient history was greatly expanded. Once Em- 
peror, Napoleon supported French science vigorously in 
an attempt to make it compete more successfully with 
British science . It was similar to the American-Soviet 
rivalry a century and a half later . 

The most famous Napoleonic tale with respect to science, 
was in connection with the astronomer Laplace, who was 
putting out the first few volumes of his Celestial Me- 
chanics which completed the work of Newton and de- 
scribed the machinery of the Solar system . Napoleon 
leafed through the book and remarked there was no men- 
tion of God . '7 had no need of that hypothesis 7* said 
Laplace. 


190 


NUMBERS AND THE CALENDAR 


Begin at the Beginning 


191 


The Romans had, traditionally, begun their year on 
March 15 (the “Ides of March”), which was intended to 
fall upon the vernal equinox originally but which, thanks 
to the sloppy way in which the Romans maintained their 
calendar, eventually moved far out of synchronization 
with the equinox. Caesar adjusted matters and moved Ihe 
beginning of the year to January 1 instead, placing it 
nearly at the winter solstice. 

This habit of beginning the year on or about the winter 
solstice did not become universal, however. In England 
(and the American colonies) March 25, intended to repre- 
sent the vernal equinox, remained the official beginning 
of the year until 1752, It was only then that the January 
1 beginning was adopted. 

The beginning of a new Sun reflects itself in modern 
times in another way, too. In the days of the Roman 
Empire, the rising power of Christianity found its most 
dangerous competitor in Mithraism, a cult that was Per- 
sian in origin and was devoted to sun worship. The ritual 
centered about the mythological character of Mithras, 
who represented the Sun, and whose birth was celebrated 
on December 25 — about the time of the winter solstice. 
This was a good time for a holiday, anyway, for the 
Romans were used to celebrating the Saturnalia at that 
time of year. 

Eventually, though, Christianity stole Mithraic thunder 
by establishing the birth of Jesus on December 25 (there 
is no biblical authority for this), so that the period of 
the winter solstice has come to mark the birth of both 
the Son and the Sun, There are some present-day moral- 
ists (of whom I am one) who find something unpleasantly 
reminiscent of the Roman Saturnalia in the modern secu- 
lar celebration of Christmas. 

But where do the years begin? It is certainly convenient 
to number the years, but where do we start the numbers? 
In ancient times, when the sense of history was not highly 
developed, it was sufficient to begin numbering the years 
with the accession of the local king or ruler. The number- 
ing would begin over again with each new king. Where n 
city has an annually chosen magistrate, the year might not 
be numbered at all, but merely identified by the name of 
the magistrate for that year. Athens named its years by its 
archons. 


When the Bible dates things at all, it does it in this man- 
ner. For instance, in II Kings 16:1, it is written: “In the 
seventeenth year of Pekah the son of Remaliah, Ahaz 
the son of Jotham king of Judah began to reign.” (Pekah 
was the contemporary king of Israel.) 

And in Luke 2:2, the time of the taxing, during which 
Jesus was born, is dated only as follows: “And this taxing 
was first made when Cyrenius was governor of Syria.” 

Unless you have accurate lists of kings and magistrates 
and know just how many years each was in power and 
how to relate the list of one region with that of another, 
you are in trouble, and it is for that reason that so many 
ancient dates are uncertain — even (as I shall soon ex- 
plain) a date as important as that of the birth of Jesus. 

A much better system would be to pick some important 
date in the past (preferably one far enough in the past 
so that you don’t have to deal with negative-numbered 
years before that time) and number of years in progres- 
sion thereafter, without ever starting over. 

The Greeks made use of the Olympian Games for that 
purpose. This was celebrated every four years so that a 
four-year cycle was an “Olympiad.” The Olympiads were 
numbered progressively, and the year itself was the 1st, 
2nd, 3rd or 4th year of a particular Olympiad. 

This is needlessly complicated, however, and in the time 
following Alexander the Great something better was 
introduced into the Greek world. The ancient East was 
being fought over by Alexander’s generals, and one of 
them, Seleucus, defeated another at Gaza. By this victory 
Seleucus was confirmed in his rule over a vast section of 
Asia. He determined to number the years from that battle, 
which took place in the 1st year of the 117th Olympiad. 
That year became Year 1 of the “Seleucid Era” and later 
years continued in succession as 2, 3, 4, 5, and so on. 
Nothing more elaborate than that. 

The Seleucid Era was of unusual importance because 
Seleucus and his descendants ruled over Judea, which 
therefore adopted the system. Even after the Jews broke 
free of the Seleucids under the leadership of the Maccabees, 
they continued to use the Seleucid Era in dating their 
commercial transactions over the length and breadth of 
the ancient world. Those commercial records can be tied 
in with various year-dating systems, so that many of them 
could be accurately synchronized as a result. 



192 


NUMBERS AND THE CALENDAR 


Begin at the Beginning 


193 


The most important year-dating system of the ancient 
world .however, was that of the ‘'Roman Era. 11 T his began 
with the vcar in which Rome was founded. According to 
tradition/ this was the 4th Year of the 6th Olympiad, 
which came to be considered as 1 a.u.c. (The abbreviation 
“a.u.c.” stands for “Anno Urbis Conditae”; that is, "The 
Year of the Founding of the City.") 

Using the Roman Era, the Battle of Zama, in which 
Hannibal was finally defeated, was fought in 553 a.u.c., 
while Julius Caesar was assassinated in 710 a.u.c., and 
so on. This system gradually spread over the ancient 
world, as Rome waxed supreme, and lasted well into early 
medieval times. 

The early Christians, anxious to show that biblical 
records antedated those of Greece and Rome, strove to 
begin counting at a date earlier than that of either the 
founding of Rome or the beginning of the Olympian 
Games. A Church historian, Eusebius of Caesarea, who 
lived about 1050 a.u.c., calculated that the Patriarch, 
Abraham, had been born 1263 years before the founding 
of Rome. Therefore he adopted that year as his Year 1, 
so that 1050 a.u.c. became 2313, Era of Abraham. 

Once the Bible was thoroughly established as the book 
of the western world, it was possible to carry matters to 
their logical extreme and date the years from the creation 
of the world. The medieval Jews calculated that the crea- 
tion of the world had taken place 3007 years before the 
founding of Rome, while various Christian calculators 
chose years varying from 3251 to 4755 years before the 
founding of Rome. These are the various “Mundane Eras” 
(“Eras of the World”). The Jewish Mundane Era is used 
today in the Jewish calendar, so that in September 1964, 
the Jewish year 5725 began. 

The Mundane Eras have one important factor in their 
favor. They start early enough so that there are very few, 
if any, dates in recorded history that have to be given 
negative numbers. This is not true of the Roman Era, for 
instance. The founding of the Olympian Games, the Trojan 
War, the reign of David . the building of the Pyramids, 
all came before the founding ol Rome and have to be 
given negative year numbers. 

The Romans wouldn't have cared, of course, for none 
of the ancients were very chronology conscious, but 


modern historians would. In fact, modern historians are 
even worse off than they would have been if the Roman 
Era had been retained. 

About 1288 A.u.c,, a Syrian monk named Dionysius 
Exiguus, working from biblical data and secular records, 
calculated that Jesus must have been born in 754 a.u.c. 
This seemed a good time to use as a beginning for count- 
ing the years, and in the time of Charlemagne (two and 
a half centuries after Dionysius) this notion won out. 

The year 754 a.u.c, became a.d. 1 (standing for Anno 
Domini, meaning “the year of the Lord”). By this new 
“Christian Era,” the founding of Rome took place in 753 
B.c. (“before Christ 1 ’). The first year of the first Olympiad 
was in 776 B.c., the first year of the Seleucid Era was in 
312 B.c., and so on. 

This is the system used today, and means that all of 
ancient history from Sumer to Augustus must be dated 
in negative numbers, and we must forever remember that 
Caesar was assassinated in 44 b.c. and that the next year 
is number 43 and not 45. 

Worse stiill, Dionysius was wrong in his calculations. 
Matthew 2:1 clearly states that “Jesus was born in Bethle- 
hem of Judea in the days of Herod the king. 1 ’ This Herod 
is the so-called Herod the Great, who was born about 
681 a.u.c., and was made king of Judea by Mark Antony 
in 714 a.u.c. He died (and this is known as certainly as 
any ancient date is known) in 750 a.u.c., and therefore 
Jesus could not have been born any later than 850 a.u.c. 

But 750 a.u.c., according to the system of Dionysius 
Exiguus, is 4 b.c., and therefore you constantly find in 
lists of dates that Jesus was born in 4 b.c.; that is, four 
years before the birth of Jesus. 

In fact, there is no reason to be sure that Jesus was 
born in the very year that Herod died. In Matthew 2:16, 
it is written that Herod, in an attempt to kill Jesus, 
ordered all male children of two years and under to be 
slain. This verse can be interpreted as indicating that 
Jesus may have been at least two years old while Herod 
was still alive, and might therefore have been born as 
early as 6 b.c. Indeed, some estimates have placed the 
birth of Jesus as early as 17 b.c. 

Which forces me to admit sadly that although I love 
to begin at the beginning, I can’t always be sure where 
the beginning is. 



194 


NUMBERS AND THE CALENDAR 


Begin at the Beginning 


195 


CHARLEMAGNE 

Charlemagne is mentioned here as the moving spirit be- 
hind the official adoption of the modern Christian era, 
which the world today almost universally uses in number- 
ing its years . 

Under Charlemagne, born in Aachen, Germany , about 
742, the Frankish Empire reached its apogee . He ruled 
over what is now France , Belgium, the Netherlands, 
Switzerland, most of Germany, most of Italy, and even 
some of Spain . The Roman Empire in the West was 
revived (after a fashion), and in 800 he was made Em- 
peror, thus beginning a tradition that was to last just over 
a thousand years and end in 1806 as a result of Napo- 
leon's conquests in Germany . 

Charlemagne’s importance in the history of science was 
that, in the midst of the period known as the Dark Age, 
he did his best to light the candles once more . He himself 
was illiterate as was almost everyone but churchmen . In 
adulthood, however, he managed to learn to read but 
could not persuade his fingers to learn to make the marks 
necessary for writing. 

He recognized the value of learning in general, too, 
and in 789 began to establish schools in which the ele- 
ments of mathematics , grammar, and ecclesiastical sub- 
jects could be taught under the over-all guidance of an 
English scholar named Alcuin . 

The result of Charlemagne's efforts is sometimes termed 
the " Carolingian renaissance ” It was a noble effort but 
a feeble one , and it did not outlast the great Emperor 
himself. He died in Aachen, January 28, 814, and he 
was succeeded by his much less talented son, Ludwig, 
usually referred to as 4f the Pious ” because he was entirely 
in the hands of the priesthood and could not control his 
family or the nobility. The coming of the Viking terror 
completed the disintegration of the abortive renaissance . 



Culver Pictures, Inc . 


Part V 


NUMBERS AND 
BIOLOGY 





THATS 
ABOUT THE 
SIZE OF IT 


No matter how much we tell ourselves that quality 
is what counts, sheer size remains impressive. The two 
most popular types of animals in any zoo are the monkeys 
and the elephants, the former because they are embar- 
rassingly like ourselves, the latter simply because they are 
huge. We laugh at the monkeys but stand in silent awe 
before the elephant. And even among the monkeys, if one 
were to place Gargantua in a cage, he would outdraw 
every other primate in the place. In fact, he did. 

This emphasis on the huge naturally makes the human 
being feel small, even puny. The fact that mankind has 
nevertheless reached a position of unparalleled domination 
of the planet is consequently presented very often as a 
David-and-Goliath saga, with ourselves as David, 

And yet this picture of ourselves is not quite accurate, 
as we can see if we view the statistics properly. 

First, let's consider the upper portion of the scale. I’ve 
just mentioned the elephant as an example of great size, 
and this is hallowed by cliche. “Big as an elephant” is a 
common phrase. 

But. of course, the elephant does not set an unqualified 
record. No land animal can be expected to. On land, an 
animal must fight gravity, undiluted. Even if it were not 
a question of lifting its bulk several feet off the ground 


199 



200 


NUMBERS AND BIOLOGY 


Thafs About the Size of It 


201 


ELEPHANTS 


The most glamorous interconnection of elephant and man 
came in ancient times when the elephant was used in 
warfare as the living equivalent of the modern tank. It 
could carry a number of men , together with important 
assault weapons. It could do damage on its own , with its 
trunk , tusks , and legs . Most of all, it represented a fear- 
ful psychological hazard to the opposing forces , who 
could only with difficulty face the giant animals . The 
greatest defects of the use of elephants lay in the fact 
that elephants were intelligent enough to run from over- 
whelming odds , and in their panic ( especially if wounded) 
could prove more damaging to their own side than to the 
enemy . 

The West came across elephants for the first time in 
326 B.c. when Alexander the Great defeated the Punjabi 
King , Porus, despite the latter s use of two hundred 
elephants. For a century afterward , the monarchs who 
succeeded Alexander used elephants . 

Usually only one side or the other had elephants, but 
at the Battle of Ipsus in 301 B.c. between rival generals 
of the late Alexander’s army , there were elephants on 
both sides , nearly three hundred in all. African elephants 
were sometimes used , though Asian elephants were more 
common. The African elephants, however, were native to 
north Africa and were smaller than the Asian elephants. 
The north African variety is now extinct , and when we 
speak of African elephants today , we mean those of 
eastern Africa, which are the giants of the species and 
the largest land mammal alive. It is the giant African 
elephant that is shown in the illustration. 

The Greek general Pyrrhus brought elephants into 
southern Italy in 280 b.c. to fight the Romans; but the 
Romans, though terrified of the beasts , fought resolutely 
anyway . The last elephant battle was that of Zama } where 
Hannibal's elephants did not help him defeat the Romans. 



The Granger Collection 


202 


NUMBERS AND BIOLOGY 


and moving it more or less rapidly, that fight sets sharp 
limits to size. If an animal were envisaged as lying flat 
on the ground, and living out its life as motionlessly as 
an oyster, it would still have to heave masses of tissue 
upward with every' breath. A beached whale dies for 
several reasons, but one is that its own weight upon its 
lungs slowly strangles it to death. 

In the water, however, buoyancy largely negates gravity, 
and a mass that would mean crushing death on land is 
supported under water without trouble. 

For that reason, the largest creatures on earth, present 
or past, are to be found among the whales. And the 
species of whale that holds the record is the blue whale 
or, as it is alternatively known, the sulfur-bottom. One 
specimen of this greatest of giants has been recorded with 
a length of 108 feet and a weight of 13114 tons.* 

Now the blue whale, like ourselves, is a mammal. If 
we want to see how we stand among the mammals, as far 
as size is concerned, let's see what the other extreme 
is like. 

The smallest mammals are the shrews, creatures that 
look superficially mouselike, but are not mice or even 
rodents. Rather, they are insectivores, and are actually 
more closely related to us than to mice. The smallest full- 
grown shrew weighs a minimum of 0.052 ounce. 

Between these two mammalian extremes stretches a 
solid phalanx of animals. Below the blue whale are other 
smaller whales, then creatures such as elephants, walruses, 
hippopotamuses, down through moose, bears, bison, horses, 
lions, wolves, beavers, rabbits, rats, mice and shrews. 
Where in this long list from largest whale to smallest shrew 
is man? 

To avoid any complications, and partly because my 
weight comes to a good, round figure of two hundred 
pounds, I will use myself as a measure.** 

Now, we can consider man either a giant or a pygmy, 
according to the frame of reference. Compared to the 
shrew he is a giant, of course, and compared to the whale 

[* This article first appeared in October 1961. For this new appearance , 

I have corrected any figures for which I have obtained better values 
since . ] 

[** /« 1964 , three years after writing this article, I lost weight and have 
kept it off ever since , / am rcow 180 pounds,] 


That's About the Size of It 203 

he is a pygmy. How do we decide which view to give the 
greater weight? 

In the first place, it is confusing to compare tons, 
pounds and ounces, let's put all three weights into a 
common unit. In order to avoid fractions (just at first, 
anyway) let's consider grams as the common unit. (For 
your reference, one ounce equals about 28.35 grams, one 
pound equals about 453.6 grams, and one ton equals 
about 907,000 grams.) 

Now, you see, we can say that a blue whale weighs as 
much as 120,000,000 grams while a shrew weighs as little 
as 1.5 grams. In between is man with a weight of 90,700 
grams. 

We are tens of thousands of grams heavier than a shrew, 
but a whale is tens of millions of grams heavier than a 
man, so we might insist that we are much more of a 
pygmy than a giant and insist on retaining the David-and- 
Goliath picture. 

But human sense and judgment do not differentiate by 
subtraction; they do so by division. The difference between 
a two-pound weight and a six-pound weight seems greater 
to us than that between a six-pound weight and a twelve 
pound weight, even though the difference is only four 
pounds in the first case and fully six pounds in the latter. 
What counts, it seems, is that six divided by two is three, 
while twelve divided by six is only two. Ratio, not differ- 
ence, is what we are after. 

Naturally, it is tedious to divide. An any fourth-grader 
and many adults will maintain, division comes under the 
heading of advanced mathematics. Therefore, it would 
be pleasant if we could obtain ratios by subtraction. 

To do this, we take the logarithm of a number, rather 
than the number itself. For instance, the most common 
form of logarithms are set up in such a fashion that 1 is 
the logarithm of 10, 2 is the logarithm of 100, 3 is the 
logarithm of 1 ,000 and so on. 

If we use the numbers themselves, we would point out 
an equality of ratio by saying that 1,000/100 is equal to 
100/10, which is division. But if we used the logarithms, 
we could point out the same equality of ratio by saying 
that 3 minus 2 is equal to 2 minus 1, which is subtraction. 

Or, again. 1,000/316 is roughly equal to 316/100. 
(Check it and see.) Since the logarithm of 1,000 is 3 and 
the logarithm of 100 is 2, we can set the logarithm of 316 



204 


NUMBERS AND BIOLOGY 


That's About the Size o} It 


205 


equal to 2.5, and then, using logarithms, we can express 
the equality of ratio by saying that 3 minus 2.5 is equal 
to 2,5 minus 2. 

So let’s give the extremes of mammalian weight in terms 
of the logarithm of the number of grams. The 120,000,- 
000-gram blue whale can be represented logarithmically 
by 8.08, while the 1.5-gram shrew is 0.18. As for the 
90,700-gram man, he is 4.96. 

As you see, man is about 4.8 logarithmic units removed 
from the shrew but only about 3.1 logarithmic units re- 
moved from the largest whale. We are therefore much 
more nearly giants than pygmies. 

In case you think all this is mathematical folderol 
and that I am pulling a fast one, what I’m saying is merely 
equivalent to this: A man is 45,000 times as massive as 
a shrew, but a blue whale is only 1,300 times as massive 
as a man. We would seem much larger to a shrew than 
a whale does to us. 

In fact, a mass that would be just intermediate between 
that of a shrew and a whale would be one with a loga- 
rithm that is the arithmetical average of 0.18 and 8.08, 
or 4.13. This logarithm represents a mass of 13,500 
grams, or 30 pounds. By that argument, a medium-sized 
mammal would be about the size of a four-year-old child, 
or a dog of moderate weight. 

Of course, you might argue that a division into two 
groups— pygmy and giant — is too simple. Why not a 
division into three groups — pygmy, moderate, and giant? 
Splitting the logarithmic range into three equal parts, we 
would have the pygmies in the range from 0.18 to 2.81, 
moderates from 2.81 to 5,44, and the giants from 5.44 
to 8.08. 

Put into common units this would mean that any ani- 
mal under 1.5 pounds would be a pygmy and any animal 
over 550 pounds would be a giant. By that line of think- 
ing, the animals between, including man, would be of 
moderate size. This seems reasonable enough, I must 
admit, and it seems a fair w T ay of showing that man, if 
not a pygmy, is also not a giant. 

But if we’re going to be fair, Jet’s be fair all the way. 
The David-and-Goliath theme is introduced with respect 
to man’s winning of overlordship on this planet; it is the 
victory of brains over brawn. But in that case, why con- 


sider the whale as the extreme of brawn? Early man 
never competed with whales. Whales stayed in the ocean 
and man stayed on land. Our battle was with land crea- 
tures only, so let’s consider land mammals in setting up 
our upper limit. 

The largest land mammal that ever existed is not alive 
today. It is the baluchitherium, an extinct giant rhinoceros 
that stood eighteen feet tall at the shoulder, and must 
have weighed in the neighborhood of 20 tons. 

As you see, the baluchitherium (which means 41 'Baluchi 
beast,” by the way, because its fossils were first found in 
Baluchistan) has less than one-seventh the mass of a blue 
whale. The logarithmic value of the baluchitherium’s mass 
in grams stands at 7.26. 

(From now on, 1 will give weights in common units 
but will follow it with a logarithmic value in parentheses. 
Please remember that this is the logarithm of the weight 
in grams every time. ) 

But, of course, the baluchitherium was extinct before 
the coming of man and there was no competition with 
him either. To make it reasonably fair, we must compare 
man with those creatures that were alive in his time and 
therefore represented potential competition. The largest 
mammals living in the time of man are the various ele- 
phants. The largest living African elephant may reach a 
total weight of 10,7 tons (6,99). To be sure, it is possible 
that man competed with still larger species of elephant 
now extinct. The largest elephant that ever existed could 
not have weighed more than 20 tons (7.26). 

(Notice, by the way, that an elephant is only about half 
as heavy as a baluchitherium and has only 5 per cent of 
the weight of a blue whale. In fact, a full-grown elephant 
of the largest living kind is only about the weight of a 
newborn blue whale.) 

Nor am I through. In battling other species for world 
domination, the direct competitors to man were other 
carnivores. An elephant is herbivorous. It might crush a 
man to death accidentally, or on purpose if angered, but 
otherwise it had no reason to harm man. A man does not 
represent food to an elephant. 

A man does represent food to a saber-toothed tiger, 
however, which, if hungry enough, would stalk, kill and 
eat a man who was only trying to stay out of the way. 
There is the competition. 



206 


NUMBERS AND BIOLOGY 


That's About the Size of It 


207 


Now the very largest animals are almost invariably 
herbivores. There are more plant calories available than 
animal calories and a vegetable diet can, on the whole, 
support larger animals than a meat diet. (Which is not to 
say that some carnivores aren't much larger than some 
herbivores.) 

To be sure, the largest animal of all, the blue whale, is 
technically a carnivore. However, he lives on tiny crea- 
tures strained out of ocean water, and this isn’t so far 
removed, in a philosophical sense, from browsing on grass. 
He is not a carnivore of the classic type, the kind with 
teeth that go snap! 

The largest true carnivore in all the history 1 of the earth 
is the sperm whale (of which Moby Dick is an example). 
A mature sperm whale, with a large mouth and a hand- 
some set of teeth in its lower jaw, may weigh seventy- 
five tons (7.83). 

But there again, we are not competing with sea crea- 
tures. The largest land carnivore among the mammals is 
the great Alaskan bear (also called the Kodiak bear), 
which occasionally tips the scale at 1,650 pounds (5.87). 
I don't know of any extinct land carnivore among the 
mammals that was larger. 

Turning to the bottom end of the scale, there we need 
make no adjustments. The shrew is a land mammal and 
a carnivore and, as far as I know, is the smallest mammal 
that ever existed. Perhaps it is the smallest mammal that 
can possibly exist. The metabolic rate of mammals goes 
up as size decreases because the surface-to-volume ratio 
goes up with decreasing size. Some small animals might 
(and do) make up for that by letting the metabolic rate 
drop and the devil with it, but a warm-blooded creature 
cannot. It must keep its temperature high, and, therefore, 
its metabolism racing (except during temporary hiberna- 
tions). 

A warm-blooded animal the size of a shrew must eat 
just about constantly to keep going. A shrew starves to 
death if it goes a couple of hours without eating; it is 
always hungry and is very vicious and ill-tempered in con- 
sequence. No one has ever seen a fat shrew or ever will. 
(And if anyone wishes to send pictures of the neighbor’s 
wife in order to refute that statement, please don't.) 

Now let’s take the range of land-living mammalian 
carnivores and break that into three parts. From 0.18 to 


2,08 are the pygmies, from 2.08 to 3.98 the moderates, 
and from 3.98 to 5.87 the giants. In common units that 
would mean that any creature under 4*4 ounces is a 
pygmy, anything from 4 1 /i ounces to 22 pounds is a mod- 
erate, and anything over 22 pounds is a giant. 

Among the mammalian land carnivores of the era in 
which man struggled through first to survival and then to 
victory, man is a giant. In the David-and-Goliath struggle, 
one of the Goliaths won. 

Of course, some suspicion may be aroused by the fact 
that I am so carefully specifying mammals throughout. 
Maybe man is only a giant among mammals, you may 
think, but were I to broaden the horizon he would turn 
out to be a pygmy after all. 

Well, not so. As a matter of fact, mammals in general 
are giants among animals. Only one kind of non-mammal 
can compete (on land) with the large mammals, and they 
are the reptile monsters of the Mesozoic era — the large 
group of animals usually referred to in common speech 
as “the dinosaurs.” 

The largest dinosaurs were almost the length of the 
very largest whales, but they were mostly thin neck and 
thin tail, so that they cannot match those same whales in 
mass. The bulkiest of the large dinosaurs, the Brachi- 
osaurus, probably weighed as much as 75 tons (7.83). 
It is the size of the sperm whale, but it is only three-fifths 
the size of the blue whale. And, as is to be expected, the 
largest of the dinosaurs were herbivores. 

The largest carnivorous dinosaurs were the allosaurs, 
some of whom may have weighed as much as twenty 
tons (7.26). An allosaur might weigh as much as a 
baluchitherium, be twice the weight of the largest elephant 
and twenty-four times the weight of the poor little Kodiak 
bear. 

The allosaurs were beyond doubt the largest and most 
fearsome land carnivores that ever lived. They and all 
their tribe, however, were gone from the earth millions 
of years before man appeared on the scene. 

If we confine ourselves to reptiles alive in man’s time, 
the largest appear to be certain giant crocodiles of South- 
east Asia. Unfortunately, reports about the size of such 
creatures always tend to concentrate on the length rather 
than the weight (this is even truer of snakes); some are 



208 


NUMBERS AND BIOLOGY 


That's About the Size of It 


209 


described as approaching thirty feet in length. I estimate 
that such monsters should also approach a maximum of 
two tons (6,25) in weight. 

I have a more precise figure for the next most massive 
group of living reptiles, the turtles. The largest turtle on 
record is a marine leatherback with a weight of 1,902 
pounds (5,93), or not quite a ton. 

To be sure, neither of these creatures is a land animal. 
The leatherback is definitely a creature of the sea, while 
crocodiles are river creatures. Nevertheless, as far as the 
crocodiles are concerned I am inclined not to omit them 
from the list of man’s competitors. Early civilizations 
developed along tropical or subtropical rivers; who is not 
aware of the menace of the crocodile of the Nile, for in- 
stance? And certainly it is a dangerous creature with a 
mouth and. teeth that go snap! to end all snaps! (What 
jungle movie would omit the terrifying glide and gape of 
the crocodile?) 

The crocodiles are smaller than the largest land-jiving 
mammals, but the largest of these reptiles would seem 
to outweigh the Kodiak bear. However, even if we let 
5.93 be the new upper limit of the “land" carnivores, 
man would still count as a giant. 

If we move to reptiles that are truly of the land, their 
inferiority to mammals in point of size is clear. The largest 
land reptile is the Galapagos tortoise, which may reach six 
hundred pounds (5.42). The largest snake is the reticu- 
lated python, .which may reach an extreme length of 
thirty-three feet. Here again, weights aren't given, as all 
the ooh’ing and ah'ing is over the measurement by yard- 
stick. However, I don’t see how this can represent a 
weight greater than 450 pounds (5.32). Finally, the larg- 
est living lizard is the Komodo monitor, which grows to 
a maximum length of twelve feet and to a weight of 250 
pounds (5.05). 

The fishes make a fairly respectable showing. The 
largest of all fishes, living or extinct, is the whale shark. 
The largest specimens of these are supposed to be as large 
and as massive as the sperm whale, though perhaps a forty- 
five-ton maximum (7.61) might be more realistic. Again, 
these sharks are harmless filterers of sea water. The 
largest carnivorous shark is the white shark, which reaches 
lengths of thirty-five feet and possibly a weight of twelve 
tons (7.03). 


Of the bony fishes, the largest (such as the tuna, sword- 
fish, sunfish, or sturgeon) may tip the scales at as much 
as three thousand pounds (6.13). All fish, however, are 
water creatures, of course, and not direct competition for 
any man not engaged in such highly specialized occupa- 
tions as pearl-diving. 

The birds, as you might expect, make a poorer show- 
ing. You can't be very heavy and still fly. 

This means that any bird that competes with man in 
weight must be flightless. The heaviest bird that ever lived 
was the flightless Aepyornis of Madagascar (also called 
the elephant bird), which stood ten feet high and may 
have weighed as much as one thousand pounds (5.66), 
The largest moas of New Zealand were even taller 
(twelve feet) but were more lightly built and did not 
weigh more than five hundred pounds (5.36). In compari- 
son, the largest living bird, the ostrich — still flightless — 
has a maximum weight of about three hundred fifty 
pounds (5.20), 

When we get to flying birds, weight drops drastically. 
The albatross has a record wingspread of twelve feet, but 
wings don’t weigh much and even the heaviest flying bird 
probably does not weigh more than forty pounds (4.26). 
Even the pteranodon, which was the largest of the extinct 
flying reptiles, and had a wingspread of up to twenty-five 
feet, was virtually all wing and no body, and probably 
weighed less than an albatross.* 

To complete the classes of the vertebrates, the largest 
amphibians are giant salamanders found in Japan, which 
are up to five feet in length and weigh up to ninety 
pounds (4,60). 

Working in the other direction, we find that the smallest 
bird, the bee hummingbird of Cuba, is about 0.07 ounce 
in weight (0.30). (Hummingbirds, like shrews, have to 
keep eating almost all the time, and starve quickly.) 

The cold-blooded vertebrates can manage smaller sizes 
than any of the warm-blooded mammals and birds, how- 
ever, since cold blood implies that body temperature can 
drop to that of the surroundings and metabolism can be 
lowered to practical levels. The smallest vertebrates of 
all are therefore certain species of fish. There is a fish of 

[* In 1975 fossils of a flying reptile much larger than the pteranodon were 
discovered. I suspect it may have weighed as much as fifty pounds .] 



210 


NUMBERS AND BIOLOGY 


That's About the Size of It 


211 


HUMMINGBIRDS 

The hummingbirds are the closest that warm-blooded 
animals can come to filling the environmental niche of 
insects . Any smaller and the capacity to produce heat by 
metabolic action could not match the loss of heat through 
the surface . 

As it is the largest hummingbird , the giant humming- 
bird is about 20 grams (0.7 ounce) in weight , which is 
rather less than that of the average sparrow , and the 
smallest is only one-tenth that size . The name of the 
smallest , the bee hummingbird , emphasizes the similarity 
to insects . It feeds on nectar and can hover in the air, 
then dart suddenly in any direction , like an outsize 
dragonfly. 

The eggs laid by the hummingbirds are the smallest of 
those laid by any bird. It would take 125 of them to 
weigh as much as a hen's egg and about 18,000 of them 
to weigh as much as the largest egg of alt , that of the 
extinct giant bird the aepyornis. Still , compared to the 
size of the hummingbird itself , the eggs are quite large . 
The two it usually lays weigh up to a tenth that of the 
mother. ( This is not a record , however . The flightless 
New Zealand bird , the kiwi , lays an egg that is almost a 
quarter of its own weight, and how it does that without 
foundering itself will always remain a puzzle to me.) 

The hummingbird is the most extravagant energy- user 
of any living organism . It expends about 103 kilocalories 
per twenty-four-hour period , which means about 5 kilo- 
calories per gram . The human being may expend 2,500 
kilocalories in the same period but that is only about 
0,035 per gram. Weight for weight , hummingbirds ex- 
pend nearly 150 times the energy we do. During night t 
however, hummingbirds become torpid, and both body 
temperature and metabolic rate drop considerably. The 
shrew, which in the article shares honors for warm- 
blooded smallness, has a slightly lower metabolic rate , 
but is as active by night as by day — no torpidity. 



The Settmann Archive 



212 


NUMBERS AND BIOLOGY 


That's About the Size of It 


213 


the goby group in the Philippine Islands that has a length, 
when full grown, of only three-eighths of an inch. Such a 
fish weighs %o of a gram (-2.70), which, as you notice, 
carries us into negative logarithms. 


What about invertebrates? 

Well, invertebrates, having no internal skeleton with 
which to brace their tissues, cannot be expected to grow 
as large as vertebrates. Only in the water, where they 
can count on buoyancy, can they make any decent show- 
ing at all. 

The largest invertebrates of all are to be found among 
the mollusks. Giant squids with lengths up to fifty-five 
feet have been actually measured, and lengths up to one 
hundred feet have been conjectured. Even so, such lengths 
are illusory, for they include the relatively light tentacles 
for the most part. The total weight of such creatures is 
not likely to be much more than two tons (6.26). 

Another type of mollusk, the giant clam, may reach a 
weight of seven hundred pounds (5.50), mostly dead 
shell, while the largest arthropod is a lobster that weighed 
in at thirty-four pounds (4,19). 

As for the land invertebrates, mass is negligible. The 
largest land crabs and land snails never match the weights 
of any but quite small mammals. The same is true of the 
most successful and important of all the land invertebrates, 
the insects. The bulkiest insect is the goliath beetle, which 
can be up to five or six inches in length, with a weight of 
about 100 grams (2.00), 

And the insects, with a top weight just overlapping the 
bottom of the mammalian scale, are well represented in 
levels of less and less massive creatures. The bottom is an 
astonishing one, for there are small beetles called fairy 
flies that are as small as l A%r> of an inch in length, full- 
grown. Such creatures can have weights of no more than 
0.000005 grams (-5.30). 

Nor is even this the record. Among the various classes 
of multicelled invertebrates, the smallest of all is Rotifera. 
Even the largest of these are only one-fifteenth of an inch 
long, while the smallest are but one three-hundredth of 
an inch long and may weigh 0.000000006 gram ( — 8.22). 
The rotifers, in other words, are to the shrews as the 
shrews are to the whales. If we go still lower, we will end 


Table 3 Sizes 


ANIMAL 

Blue whale 
Sperm whale 
Brachiosaurus 

Whale shark 
Allosaur 

Baluchitherium 

White shark 
Elephant 

Giant squid 
Crocodile 
Sunfish 
Leatherback 
Kodiak bear 

Aepyornis 
Giant clam 
Galapagos tortoise 

Reticulated python 
Ostrich 

Komodo monitor 
Man 

Giant salamander 

Albatross 

Lobster 

Goliath beetle 

Bee hummingbird 

Shrew 

Goby 

Fairy fly 
Rotifer 



LOGARITHM 


OF WEIGHT 

CHARACTERISTIC 

IN GRAMS 

Largest of all animals 

8.08 

Largest of all carnivores 

7.83 

Largest land animal 


(extinct) 

7.83 

Largest fish 

7.61 

Largest land carnivore 


(extinct) 

7.26 

Largest land mammal 


(extinct) 

7.26 

Largest carnivorous fish 

7.03 

Largest land animal 


(alive) 

6.99 

Largest invertebrate 

6.26 

Largest reptile (alive) 

6.25 

Largest bony fish 

6.13 

Largest turtle 

5.93 

Largest land carnivore 


(alive) 

5.87 

Largest bird (extinct) 

5.66 

Largest gastropod 

5.50 

Largest land reptile 


(alive) 

5.42 

Largest snake 

5.32 

Largest bird (alive) 

5.20 

Largest lizard 

5.05 


4.96 

Largest amphibian 

4.60 

Largest flying bird 

4.26 

Largest arthropod 

4.19 

Largest insect 

2.00 

Smallest bird 

0.30 

Smallest mammal 

0.18 

Smallest fish and 


vertebrate 

—2,70 

Smallest insect 

-5.30 

Smallest multicelled 


creature 

— 8,22 



214 


NUMBERS AND BIOLOGY 


considering not only man but also the shrew as a giant 
among living creatures. 

But below the rotifers are the one-celled creatures 
(though, in fact, the larger one-ceiled creatures are larger 
than the smallest rotifers and insects), and I will stop 
here, adding only a summarizing table of sizes. 

But if we are to go back to the picture of David and 
Goliath, and consider man a Goliath, we have some real 
Davids to consider — rodents, insects, bacteria, viruses. 
Come to think of it, the returns aren’t yet in, and the 
wise money might be on the real Davids after all. 


Part VI 


NUMBERS AND 
ASTRONOMY 



M the 

PROTON- 

RECKONER 


There is, in my heart, a very warm niche for the 
mathematician Archimedes. 

In fact, if transmigrations of souls were something 1 
believed in, I could only wish that my soul had once 
inhabited the body of Archimedes, because I feel it would 
have had a congenial home there. 

IT1 explain why. 

Archimedes was a Greek who lived in Syracuse, Sicily* 
He was born about 287 b.c. and he died in 212 b.c. His 
lifetime covered a period during which the great days of 
Greece (speaking militarily and politically) were long 
since over, and when Rome was passing through its 
meteoric rise to world power. In fact, Archimedes died 
during the looting of Syracuse by the conquering Roman 
army. The period, however, represents the century during 
which Greek science reached its height — and Archimedes 
stands at the pinnacle of Greek science. 

But that's not why I feel the particular kinship with 
him (after all, I stand at no pinnacle of any science). It 
is rather because of a single work of his; one called 
"Psammites” in Greek, “Arenarius” in Latin, and “The 
Sand-Reckoner” in English. 

It is addressed to Gelon, the eldest son of the Syra- 
cusan king, and it begins as follows: 


217 



218 


NUMBERS AND ASTRONOMY 


The Proton-Reckoner 


219 


“There are some, king Gelon, who think that the num- 
ber of the sand is infinite in multitude; and I mean by 
the sand not only that which exists about Syracuse and 
the rest of Sicily but also that which is found in every 
region whether inhabited or uninhabited. Again there are 
some who, without regarding it as infinite, yet think that 
no number has been named which is great enough to 
exceed its multitude. And it is clear that they who hold 
this view, if they imagined a mass made up of sand in 
other respects as large as the mass of the earth, including 
in it all the seas and the hollows of the earth filled up to 
a height equal to that of the highest of the mountains, 
would be many times further still from recognizing that 
any number could be expressed which exceeded the multi- 
tude of the sand so taken. But I will try to show you by 
means of geometrical proofs, which you will be able to 
follow, that of the numbers named by me and given in the 
work which I sent to Zeuxippus, some exceed not only 
the number of the mass of sand equal in magnitude to 
the earth filled up in the way described but also that of a 
mass equal in magnitude to the universe.” 

Archimedes then goes on to invent a system for ex- 
pressing large numbers and follows that system dear up 
to a number which we would express as lO 30 - 000000 - 000 ' 000 ' 000 , 
or nearly 10 101 \ 

After that, he sets about estimating the size of the 
universe according to the best knowledge of his day. He 
also sets about defining the size of a grain of sand. Ten 
thousand grains of sand, he says, would be contained in 
a poppy seed, where the poppy seed is *4o of a finger- 
breadth in diameter. 

Given the size of the universe and the size of a grain 
of sand, he easily determines how many grains of sand 
would be required to fill the universe. It works out to a 
certain figure in his system of numbers, which in our 
system of numbers is equal to 1 0 G3 . 

It’s obvious to me (and I say this with all possible 
respect) that Archimedes was writing one of my science 
essays for me, and that is why he has wormed his way into 
my heart. 

But let’s see what can be done to advance his article 
further in as close an approach as possible to the original 
spirit 


The diameter of a poppy seed, says Archimedes, is 14o 
of a finger-breadth. My own fingers seem to be about 20 
millimeters in diameter and so the diameter of a poppy 
seed would be, by Archimedes’ definition, 0.5 millimeter. 

If a sphere 0.5 millimeter in diameter will hold 10,000 
(10 4 ) grains of sand and if Archimedes’ universe will 
hold 10 63 grains of sand, then the volume of Archimedes’ 
universe is 10 59 times as great as that of a poppy seed. 

The diameter of the universe would then bey^O 59 times 
as great as that of a poppy seed. The cube root of 10 39 
is equal to 4.65 xlO 19 and if that is multiplied by 0.5 
millimeter, it turns out that Archimedes’ universe is 
2.3 XlO 19 millimeters in diameter, or, taking half that 
value, 1.15 XlO 19 millimeters in radius. 

This radius comes out to 1.2 light-years. In those days, 
the stars were assumed to be fixed to a large sphere with 
the Earth at the center, so that Archimedes was saying 
that the sphere of the fixed stars was about 1.2 light- 
years from the Earth in every direction. 

This is a very respectable figure for an ancient mathe- 
matician to arrive at, at a time when the true distance of 
the very nearest heavenly body — the Moon — was just in 
the process of being worked out and when all other dis- 
tances were completely unknown. 

Nevertheless, it falls far short of the truth and even 
the nearest star, as we now know, is nearly four times 
the distance from us that Archimedes conceived all the 
stars to be. 

What, then, is the real size of the universe? 

The objects in the universe which are farthest from us 
are the galaxies; and some of them are much farther 
than others. Early in the twentieth century, it was deter- 
mined that the galaxies (with a very few exceptions among 
those closest to us) were all receding from us. Further- 
more, the dimmer the galaxy and therefore the farther 
(presumably), the greater the rate of recession. 

In 1929, the American astronomer Edwin Powell Hubble 
decided that, from the data available, it would seem that 
there was a linear relationship between speed of recession 
and distance. In other words, if galaxy 1 were twice as far 
as galaxy 2, then galaxy 1 would be receding from us at 
twice the velocity that galaxy 2 would be. 



220 


NUMBERS AND ASTRONOMY 


The Proton-Reckoner 


221 


This relationship (usually called Hubble's Law) can 
be expressed as follows: 

R=kD (Equation 1) 

where R is the speed of recession of a galaxy, D its dis- 
tance, and k a constant, which we may call “Hubble’s 
constant,” 

This is not one of the great basic laws of the universe 
in which scientists can feel complete confidence. However, 
in the nearly forty years since Hubble’s Law was pro- 
pounded, it does not seem to have misled astronomers and 
no observational evidence as to its falsity has been ad- 
vanced. Therefore, it continues to be accepted.* 

One of the strengths of Hubble’s Law is that it is the 
sort of thing that w T ould indeed be expected if the universe 
as a whole (but not the matter that made it up) were 
expanding. In that case, every galaxy would be moving 
away from every other galaxy and from the vantage point 
of any one galaxy, the speed of recession of the other 
galaxies would indeed increase linearly with distance. 
Since the equations of Einstein’s General Theory of Rela- 
tivity can be made to fit the expanding universe (indeed 
the Dutch astronomer Willem de Sitter suggested an ex- 
panding universe years before Hubble’s Law was pro- 
posed) astronomers are reasonably happy. 

But what is the value of Hubble’s constant? The first 
suggestion was that it was equal to five hundred kilo- 
meters per second per million parsecs. That would mean 
that an object a million parsecs away would be receding 
from us at a speed of five hundred kilometers per second; 
an object two million parsecs away at a speed of one 
thousand kilometers per second; an object three million 
parsecs away at a speed of fifteen hundred kilometers per 
second, and so on. 

This value of the constant, it turned out, was too high 
by a considerable amount. Current thinking apparently 
would make its value somewhere between seventy-five and 
one hundred and seventy-five kilometers per second per 
million parsecs. Since the size of the constant has been 

[* This article first appeared in January 1966 . In the decade since , there 
has been considerable argument over Hubble's Law, and while it is still 
accepted by astronomers, my attitude would not be as complacent now 
as it was then.] 


shrinking as astronomers gain more and more information, 
I suspect that the lower limit of the current estimate is 
the most nearly valid value and I will take seventy-five 
kilometers per second per million parsecs as the value 
of Hubble’s constant. 

In that case, how far distant can galaxies exist? If, with 
every million parsecs, the speed of recession increases by 
seventy-five kilometers per second, then, eventually, a 
recession equal to the speed of light (three hundred thou- 
sand kilometers per second) will be reached. 

And what about galaxies still more distant? If Hubble’s 
Law holds firmly at all distances and if we ignore the 
laws of relativity, then galaxies still farther than those 
already receding at the speed of light must be viewed as 
receding at speeds greater than that of light. 

We needn’t pause here to take up the question as to 
whether speeds greater than that of light are possible or 
not, and whether such beyond-the-limit galaxies can exist 
or not. It doesn’t matter. Light from a galaxy receding from 
us at a speed greater than light cannot reach us; nor can 
neutrinos nor gravitational influence nor electromagnetic 
fields nor anything. Such galaxies cannot be observed in 
any way and therefore, as far as we are concerned, do 
not exist, whether we argue according to the Gospel of 
Einstein or the Gospel of Newton. 

We have, then, what we call an Observable Universe. 
This is not merely that portion of the universe which 
happens to be observable with our best and most powerful 
instruments; but that portion of the universe which is all 
that can be observed even with perfect instruments of 
infinite power. 

The Observable Universe, then, is finite in volume and 
its radius is equal to that distance at which the speed of 
recession of a galaxy is three hundred thousand kilometers 
per second. 

Suppose we express Equation 1 as 

D=R/k (Equation 2) 

set R equal to three hundred thousand kilometers per 
second and k equal to seventy-five kilometers per second 
per million parsecs. We can then solve for D and the 
answer will come out in units of million parsecs. 



222 NUMBERS AND ASTRONOMY 

It turns out, then, that 

30,000 -75=4,000 (Equation 3) 

The farthest possible distance from us; or, what amounts 
to the same thing, the radius of the Observable Universe; 
is 4,000 million parsecs, or 4,000,000,000 parsecs. A 
parsec is equal to 3.26 light-years, which means that the 
radius of the Observable Universe is 13,000,000,000 light- 
years. This can be called the Hubble Radius. 

Astronomers have not yet penetrated the full distance 
of the Hubble Radius, but they are approaching it. From 
Mount Palomar comes word that the astronomer Maarten 
Schmidt has determined that an object identified as 3C9 
is receding at a speed of 240,000 kilometers per second, 
four-fifths the speed of light. That object is, therefore, a 
little more than ten billion light-years distant, and is the 
most distant object known. * 

As you see, the radius of the Observable Universe is 
immensely greater than the radius of Archimedes' uni- 
verse: thirteen billion as compared to 1.2. The ratio is 
just about ten billion. If the volumes of two spheres are 
compared, they vary with the cube of the radius. If the 
radius of the Observable Universe is 10 10 times that of 
Archimedes’ universe, the volume of the former is 
C 10 10 ) 3 or IQ 30 times that of the latter. 

If the number of sand particles that filled Archimedes’ 
universe is I0 G3 , then the number required to fill the im- 
mensely larger volume of the Observable Universe is 10 &3 . 

But, after all, why cling to sand grains? Archimedes 
simply used them in order to fill the greatest possible 
volume with the smallest possible objects, Indeed, he 
stretched things a little. If a poppy seed 0.5 millimeter 
in diameter will hold ten thousand grains of sand, then 
each grain of sand must be 0.025 millimeter in diam- 
eter. These are pretty fine grains of sand, individually 
invisible to the eye. 

We can do better. We know of atoms, which Archi- 
medes did not, and of subatomic particles, too. Suppose 
we try to search among such objects for the smallest 


[* hi 1973 an object at a distance of twelve billion light-years was 
detected 4 ] 


The Proton-Reckoner 223 

possible volume; not merely a volume, but the Smallest 
Possible Volume. 

If it were the smallest possible mass we were searching 
for, there would be no problem; it would be the rest mass 
of the electron which is 9.1x10-*® grams. No object that 
has any mass at all has a smaller mass than the electron. 
(The positron has a mass that is as small, but the positron 
is merely the electron’s anti-particle, the looking-glass 
version of the electron, in other words.) 

There are particles less massive than the electron. 
Examples are the photon and the various neutrinos, but 
these all have zero rest mass, and do not qualify as an 
“object that has any mass at all.” 

Why is this? Well, the electron has one other item of 
uniqueness. It is the least massive object which can carry 
an electric charge. Particles with zero rest mass are in- 
variably electrically uncharged, so that the existence of 
electric charge seems to require the presence of mass — 
and of mass no smaller than that associated with the 
electron. 

Perhaps electric charge is mass, and the electron is 
nothing but electric charge — whatever that is. 

Yet it is possible to have a particle such as the proton, 
which is 1,836 times as massive as the electron, with an 
electric charge no greater. Or we can have a particle 
such as a neutron, which is 1,838 times as massive as an 
electron and has no charge at all. 

We might look at such massive, undercharged particles 
as consisting of numerous charges of both types, positive 
and negative, most or all of which cancel one another, 
leaving one positive charge in excess in the case of the 
proton, and no uncanceled charge at all in the case of 
the neutron. 

But, then, how can charges cancel each other without, 
at the same time, cancelling the associated mass? No one 
knows. The answer to such questions may not come 
before considerably more is learned about the internal 
structure of protons and neutrons. We will have to wait. 

Now what about volume? 

We can talk about the mass of subatomic particles with 
confidence, but volume is another matter. All particles 
exhibit wave properties, and associated with all chunks of 
matter are “matter-waves” of wavelength varying inversely 



224 


NUMBERS AND ASTRONOMY 


The Proton-Reckoner 


225 


with the momentum of the particles (that is, with the 
product of their mass and velocity) . 

The matter-waves associated with electrons have wave- 
lengths of the order of 10“* centimeters, which is about 
the diameter of an atom. It is therefore unrealistic to talk 
about the electron as a particle, or to view it as a hard, 
shiny sphere with a definite volume. Thanky to its wave 
nature, the electron “smears out’ 5 to fill the atom of which 
it forms a part. Sometimes it is “smeared out” over a 
whole group of atoms. 

Massless particles such as photons and neutrinos are 
even more noticeably wave-forms in nature and can even 
less be spoken of as having a volume. 

If we move on to a proton, however (or a neutron), 
we find an object with a mass nearly two thousand times 
an electron. This means that all other things being equal 
the wavelength of the matter-wave associated with the 
proton ought to be about a two-thousandth that associated 
with the electron. 

The matter-waves are drawn in tightly about the proton 
and its particulate nature is correspondingly enhanced. 
The proton can be thought of as a particle and one can 
speak of it as having a definite volume, one that is much 
less than the wavelength of the smeared out electron. 
(To be sure, if a proton could be magnified sufficiently 
to look at we would find it had a hazy surface with no 
clear boundary so that its volume would be only approxi- 
mately “definite.”) 

Suppose we pass on to objects even more massive than 
the proton. Would the matter-waves be drawn in still 
farther and the volume be even less? There are subatomic 
particles more massive than the proton. All are extremely 
short-lived, however, and I have come across no estimates 
of their volumes. 

Still, we can build up conglomerations of many protons 
and neutrons which are stable enough to be studied. These 
are the various atomic nuclei. An atomic nucleus built up 
of, say, ten protons and ten neutrons would be twenty 
times as massive as a single proton and the matter-waves 
ascociated with the nucleus as a whole would have a 
wavelength correspondingly shorter. Would this contract 
the volume of the twenty protons and neutrons to less 
than that of a single proton? 

Apparently not. By the time you reach a body as 


massive as the proton, its particular nature is so promi- 
nent that it can be treated almost as a tiny billiard ball. 
No matter how many protons and neutrons are lumped 
together in an atomic nucleus each individual proton and 
neutron retains about its original volume. This means 
that the volume of a proton may well be considered as the 
smallest volume that has any meaning. That is, you can 
speak of a volume “half that of a proton” but you will 
never find anything that will fill that volume without 
lapping oven either as a particle or a wave. 

The sizes of various atomic nuclei have been calculated. 
The radius of a carbon nucleus, for instance, has been 
worked out as 3.8 x 10 -13 centimeters and that of a bismuth 
nucleus as about SxlO 13 centimeters. If a nucleus is 
made up of a closely packed sphere of incompressible 
neutrons or protons then the volume of two such spheres 
ought to be related as the cube roots of the number of 
particles. The number of particles in a carbon nucleus is 
12 (6 protons and 6 neutrons) and the number in a bis- 
muth nucleus is 209 (83 protons and 126 neutrons). The 
ratio of the number of particles is 209/12 or 17.4 and 
the cube root of 17.4 is 2.58. Therefore, the radius of 
the bismuth nucleus should be 2.58 times that of the 
carbon nucleus and the actual ratio is 2.1. In view of the 
uncertainties of measurement, this isn’t bad. 

Let’s next compare the carbon neucleus to a single pro- 
ton (or neutron). The carbon nucleus has twelve particles 
and the proton but one. The ratio is 12 and the cube 
root of that is just about 2.3. Therefore, the radius of the 
carbon nucleus ought to be about 2.3 times the radius 
of a proton. We find, then, that the radius of a proton 
is about 1.6x10 ]3 centimeters. 

Now we can line up protons side by side and see how 
many will stretch clear across the Observable Universe. 
If we divide the radius of the Observable Universe by 
the radius of a proton, we will get the answer. 

The radius of the Observable Universe is thirteen bil- 
lion light-years or 1.3xl0 l ° light-years, and each light- 
year is 9.5 XlO 17 centimeters long. In centimeters then, 
the radius of the Observable Universe is 1.23 x 10 28 . Divide 
that by the radius of the proton, which is 1.6 xlO -13 
centimeters and you have the answer: 77xl0 40 . 

In other words, if anyone ever asks you: “How many 



226 


NUMBERS AND ASTRONOMY 


The Proton-Reckoner 


227 


protons can you line up side by side? 1 ’ you can an- 
swer "77,000,000,000,000,000,000,000,000,000,000,000,- 
000,000!” because there is no room to line up any more. 

Now for volume. If the proton has a radius of 1.6 X 10~ 13 
centimeters and it is assumed to be spherical, it has a 
volume of 1.7x10 40 cubic centimeters, and that is the 
Smallest Possible Volume. Again, given a radius of 
1.23 XlO 28 centimeters for the Observable Universe, its 
volume is 7.8 XlO 84 cubic centimeters, and that is the 
Greatest Possible Volume. 

We next suppose that the Greatest Possible Volume is 
packed perfectly tightly (leaving no empty spaces) with 
objects of the Smallest Possible Volume. If we divided 
7.8 XlO 84 by 1.7 XlO' 49 , we find that the number of pro- 
tom it takes to fill the Observable Universe is 4.6 xl0 m . 

That is the solution (by modern standards) of the 
problem that Archimedes proposed for himself in "The 
Sand-Reckoner" and, oddly enough, the modern solution 
is almost exactly the square of Archimedes 1 solution. 

However, Archimedes need not be abashed at this, 
wherever he may be along the Great Blackboard in the 
Sky. He was doing more than merely chopping figures to 
come up with a large one. He was engaged in demonstrat- 
ing an important point in mathematics; that a number 
system can be devised capable of expressing any finite 
number however large; and this he succeeded perfectly 
in doing. 

Ah, but I’m not quite done. How many protons are 
there really in the Observable Universe? 

The "cosmic density” — that is, the quantity of matter 
in the universe, if all of it were spread out perfectly 
evenly — has been estimated at figures ranging from 10' 30 
to 10 £9 grams per cubic centimeter. This represents a 
high-grade vacuum which shows that there is practically 
no matter in the universe. Nevertheless, there are an enor- 
mous number of cubic centimeters in the universe and 
even "practically no matter” mounts up. 

The volume of the Observable Universe, as 1 said, is 
7.8 XlO 84 cubic centimeters and if the cosmic density is 
equal throughout the universe and not merely in the few 
billion light-years nearest ourselves, then the total mass 
contained in the Observable Universe is from 7.8 XlO 54 
grams to 7.8 xlO 53 grams, Let’s hit that in between and 


say that the mass of the Observable Universe is 3 x 10 55 
grams. Since the mass of our own Milky Way Galaxy 
is about 3xI0 44 grams, there is enough mass in the Ob- 
servable Universe to make up a hundred billion ( 1 0 1 1 ) 
galaxies like our own. 

Virtually all this mass is resident in the nucleons of the 
Universe, i,e the protons and neutrons. The mass of 
the individual proton or neutron is about 1.67xlO~ 24 , 
which means that there are something like 1.8 XlO 79 
nucleons in the Observable Universe. 

As a first approximation we can suppose the universe to 
be made up of hydrogen and helium only, with ten atoms 
of the former for each atom of the latter. The nucleus 
of the hydrogen atom consists of a single proton and the 
helium atom consists of two protons and two neutrons. 
In every eleven atoms, then, there is a total of twelve 
protons and two neutrons. The ratio of protons to neutrons 
in the universe is therefore six to one or roughly 1.6 X 10 7D 
protons and 0.2 XlO 70 neutrons in the universe. (There 
are thus ten quadrillion times as many protons in the 
nearly empty Observable Universe as there are sand grains 
in Archimedes’ fully packed universe.) 

In addition, each proton is associated with an electron, 
so that the total number of particles in the Observable 
Universe (assuming that only protons, neutrons, and elec- 
trons exist in significant numbers) is 3.4x10™. 

This proton-reckoning in the Observable Universe ig- 
nores relativistic effects. The farther away a galaxy is and 
the more rapidly it recedes from us, the greater the fore- 
shortening it endures because of the Fitzgerald contraction 
(at least to our own observing eyes). 

Suppose a galaxy were at a distance of ten billion 
light-years and were receding from us at four-fifths the 
speed of light. Suppose, further, we saw it edge-on so 
that ordinarily its extreme length in the line of sight 
would be one hundred thousand light-years. Because of 
foreshortening, we would observe that length (assuming 
we could observe it) to be only sixty thousand light-years. 

Galaxies still farther away would seem even more fore- 
shortened, and as we approached the Hubble Radius of 
thirteen billion light-years, where the speed of recession 
approaches the speed of light, that foreshortening would 
make the thickness of the galaxies in the line of sight 



228 NUMBERS AND ASTRONOMY 

approach zero. We have the picture then, of the neighbor- 
hood of the Hubble Radius occupied by paper-thin and 
paper-thinner galaxies. There would be room for an in- 
finite number of them, all crowded against the Hubble 
Radius. 

Inhabitants of those galaxies would see nothing wrong, 
of course. They and their neighbors would be normal 
galaxies and space about them would be nearly empty. 
But at their Hubble Radius there would be an infinite 
number of paper-thinner galaxies, including our own! 

It is possible, then, that within the finite volume of a 
nearly empty universe, there is — paradoxical though it 
may sound — an infinite universe after all, with an infinite 
number of galaxies, an infinite mass, and, to get back to 
the central point of this article, an infinite number of 
protons. 

Such a picture of an infinite universe in a finite volume 
does not square with the “big bang'* theory of the uni- 
verse, which presupposes a finite quantity of mass to begin 
with; but it fits the '‘continuous creation” universe which 
needs an infinite universe, however finite the volume. 

The weight of observation is inclining astronomers 
more and more to the “big bang” but I find myself emo- 
tionally attracted to the optimistic picture of “continuous 
creation.” 

So far we can only penetrate ten billion years into 
space, but I wait eagerly. Perhaps in my lifetime, we can 
make the final three billion light-years to the edge of the 
Observable Universe and get some indication, somehow, 
of the presence of an infinite number of galaxies there. 

But perhaps not. The faster the galaxies recede, the less 
energy reaches us from them and the harder they are to 
detect. The paper-thin galaxies may be there — but may be 
indetectable. 

If the results are inconclusive, I will be left with nothing 
but faith. And my faith is this — that the universe is 
boundless and without limit and that never, never, never 
will mankind lack for a frontier to face and conquer.* 


Part VII 


NUMBERS AND 
THE EARTH 


[* A las, in the years since this article first appeared, “ continuous crea- 
tion' 1 Has about vanished and no signs of accumulating galaxies toward 
the rim have appeared — but I retain my faith*'} 




WATER, 

WATER, 

EVERYWHERE- 


The one time in my adult life that I indulged in an 
ocean voyage, it wasn’t voluntary.* Some nice sergeants 
were herding a variety of young men in soldier suits onto 
a vessel and I was one of the young men. 

I didn’t really want to leave land (being a lubber of the 
most fearful variety) and meant to tell the sergeants so. 
However, they seemed so careworn with their arduous 
duties, so melancholy at having to undertake the uncon- 
genial task of telling other people what to do, that I 
didn’t have the heart. I was afraid that if they found 
out one of the soldiers didn’t really want to go, they 
might cry. 

So I went aboard and we began the long six-day ocean 
voyage from San Francisco to Hawaii. 

A luxury cruise it was not. The bunks were stacked 
four high and so were the soldiers. Seasickness was ram- 
pant and while I myself was not seasick even once (on 
my honor as a science fiction writer) that doesn’t mean 
much when the guy in the bunk above decides to be. 

My most grievous shock came the first night. I had 
been withstanding the swaying of the ship all day and 


[* This article first appeared in December 1965, Since then I Have been 
on a number of ocean voyages, every one of them voluntary and every 
one of them enjoyable. ] 


231 



232 


NUMBERS AND THE EARTH 


Water, Water , Everywhere — 


233 


had waited patiently for bedtime. Bedtime came; I got 
into my nonluxurious bunk and suddenly realized that 
they didn't turn off the ocean at night! The boat kept 
swaying, pitching, yawing, heaving, rolling, and other- 
wise making a jackass of itself all night long! And every 
night! 

You may well imagine, then, that what with one thing 
and another, I made the cruise in grim-lipped silence and 
was notable above all other men on board ship for my 
surly disposition. 

Except once. On the third day out, it rained. Nothing 
remarkable, you think? Remember, Tm a landlubber. 
I had never seen it rain on the ocean; I had never thought 
of rain on the ocean. And now I saw it — a complete 
waste of effort. Tons of water hurling down for nothing; 
just landing in more water. 

The thought of the futility of it all; of the inefficiency 
and sheer ridiculousness of a planetary design that allowed 
rain upon the ocean struck me so forcibly that I burst 
into laughter. The laughter fed upon itself and in no time 
at all, I was down on the deck, howling madly and 
flailing my arms and legs in wild glee — and getting rained 
on, 

A sergeant (or somebody) approached and said, with 
warm and kindly sympathy, “What the hell’s the matter 
with you, soldier? On your feet!” 

And all I could say was, “It’s raining! It’s raining on 
the ocean!” 

I kept on tittering about it all day, and that night all 
the bunks in the immediate neighborhood of mine were 
empty. The word had gone about (I imagine) that I 
was mad, and might turn homicidal at any moment. 

But many times since, I have realized I shouldn’t have 
laughed. I should have cried. 

We here in the northeastern states are suffering from a 
serious drought * and when I think of all the rain on the 
ocean and how nicely we could use a little bit of that rain 
on particular portions of dry land, I could cry right now. 

I’ll console myself as best I may, then, by talking 
about water. 

Actually, the Earth is not short of water and never will 


[* This it’as 1965, remember. Since then (knock wood) no droughts,] 


be. In fact, we are in serious and continuing danger of 
too much water, if the warming trend continues and the 
ice caps melt. * 

But let’s not worry about melting ice caps right now; 
let’s just consider the Earth’s water supply. To begin 
with, there is the ocean, I use the singular form of the 
noun because there is actually only one World Ocean; a 
continuous sheet of salt water in which the continents 
are set as large islands. 

The total surface area of the World Ocean is 139,480,- 
000 square miles, while the surface area of the entire 
planet is 196,950,000 square miles.** As you see, then, 
the World Ocean covers seventy-one per cent of the 
Earth’s surface. 

The World Ocean is arbitrarily divided into smaller 
units partly because, in the early age of exploration, men 
weren’t sure that there was a single ocean (this was first 
clearly demonstrated by the circumnavigation of the Earth 
by Magellan’s expedition in 1519-1522) and partly be- 
cause the continents do break up the World Ocean into 
joined segments which it is convenient to label separately. 

Traditionally, one hears of the “Seven Seas” and indeed 
my globe and my various atlases do break up the World 
Ocean into seven subdivisions; 1) North Pacific, 2) South 
Pacific, 3) North Atlantic, 4) South Atlantic, 5) Indian, 
6) Arctic, 7) Antarctic. 

In addition, there are the smaller seas and bays and 
gulfs; portions of the ocean which are nearly surrounded 
by land as in the case of the Mediterranean Sea or the 
Gulf of Mexico or marked off from the main body of the 
ocean by a line of islands, as in the case of the Caribbean 
Sea or the South China Sea. 

Let’s simplify this arrangement as far as possible. In 
the first place, let’s consider all seas, bays, and gulfs to 
be part of the ocean they adjoin. We can count the 
Mediterranean Sea, the Gulf of Mexico, and the Carib- 
bean Sea as part of the North Atlantic, while the South 
China Sea is part of the North Pacific. 


[• J Mi a little behind here. Actually we've been experiencing a cooling 
trend s ince 1940,] 

[** U 1 were writing the article today I would use “ square kilometers' r 
my unit , but it would be tedious to make the change now, fust remem- 
ber that J square mile equals 2.6 square kilometers and you can make the 
change yourself, if you wish,] 



234 


NUMBERS AND THE EARTH 


THE OCEAN 

As mentioned in the article f the ocean covers 71 per cent 
of the Earth f s surface. But what we see is, of course, only 
the top of it . 

On the average, the ocean is 2.3 miles (3.7 kilometers) 
deep, and there are places where it is over 7 miles (11 
kilometers) deep. The total volume of the ocean is about 
300 million cubic miles (1,200 million cubic kilometers). 
That means if you built a square tank 36 miles (58 kilo- 
meters) on each side and poured all the ocean water into 
it, you would have to build the walls as high as the Moon 
in order to hold it all . 

Ocean water is not pure water , but is a solution of 
various substances, chiefly salt . // is, in fact , 3.45 per 
cent solids, mostly salt , This means that there are some 
54,000 trillion tons of solids dissolved in the ocean , and 
if all of this could somehow be removed and spread out 
evenly over the fifty states, it would make a heap IV 2 
miles ( 2 V 2 kilometers) high. 

The sohds contained in the ocean are not exclusively 
salt. About one seventh of the solids include substances 
containing every element on Earth — some present in 
greater quantity , some in lesser. The ocean contains as 
part of its normal content even such substances as ura- 
nium and gold. In every ton of ocean water there is about 
one ten-thousandth ounce of uranium and about one five- 
millionth ounce of gold. The uranium and gold are spread 
out so thinly that it isn*t practical to try to concentrate 
the metals and extract them from the water. Still, the ocean 
is so huge that the total amount present is great. The ocean 
contains a total of 5 billion tons of uranium and 8 million 
tons of gold. 

The ocean contains dissolved gases as well Oxygen dis- 
solves in water only slightly, but there is enough dissolved 
oxygen in the ocean to support its entire load of life . 


236 


NUMBERS AND THE EARTH 


Water , Water. Everywhere— 


237 


Second, there is no geophysical point in separating the 
North Pacific from the South Pacific, or the North Atlantic 
from the South Atlantic. (The conventional arbitrary 
dividing line in each case is the equator). Let’s deal with 
a single Pacific Ocean and a single Atlantic Ocean, 

Third, if you will look at a globe, you will see that the 
Arctic Ocean is not a truly separate ocean. It is an off- 
shoot of the Atlantic Ocean to which it is connected by 
a thousand-xnile-wide passage (the Norwegian Sea) be- 
tween Greenland and Norway, Let’s add the Arctic to the 
Atlantic, therefore. 

Fourth, there is no Antarctic Ocean, The name is given 
to the stretch of waters neighboring Antarctica (which is 
the only portion of the globe where one can circumnavi- 
gate the planet along a parallel of latitude without being 
obstructed by land or by solid ice sheets). However, there 
are no nonarbitrary boundaries between this stretch of 
water and the larger oceans to the north. The length of 
arbitrary boundary can be shortened by dividing the 
Antarctic among those larger oceans. 

That leaves us, then, with exactly three large divisions of 
the World Ocean— the Pacific Ocean, the Atlantic Ocean, 
and the Indian Ocean. 

If you look at a globe, you will see that the Pacific 
Ocean and Atlantic Ocean stretch from north polar re- 
gions to south polar regions. The division between them 
in the north is clearcut, since the only connection is 
through the narrow Bering Strait between Alaska and 
Siberia. A short arbitrary line, fifty-six miles in length, 
can be drawn across the stretch of water to separate the 
oceans. 

In the south the division is less clear cut. An arbitrary 
line must be drawn across Drake Passage from the south- 
ernmost point of South America to the northernmost 
point of the Antarctic Peninsula. This line is about six 
hundred miles long. 

The Indian Ocean is the stubby one, stretching only 
from the tropics to the Antarctic Ocean (though it makes 
up for that by being wider than the Atlantic, which is the 
skinny one.) The Indian Ocean is less conveniently sepa- 
rated from the other oceans. A north-south line from the 
southernmost points of Africa and Australia to Antaclica 
will separate the Indian Ocean from the Atlantic and 
Pacific respectively. The first of these lines is about 


twenty-five hundred miles long and the second eighteen 
hundred miles long, which makes the demarcation pretty 
vague, but then I told you there’s really only one ocean. 
In addition, the Indonesian islands separate the Pacific 
from the Indian. 

The surface areas of the three oceans, using these con- 
ventions, are expressed in Table 4 in round figures: 

Table 4 Area of the Oceans 


Pacific 

Atlantic 

Indian 


SURFACE AREA 
(SQUARE MILES) 
68,000,000 
41,500,000 
30,000.000 


PER CENT OF 
WORLD OCEAN 

48.7 

29.8 
21.5 


As you see, the Pacific Ocean is as large as the Atlantic 
and Indian put together. The Pacific Ocean is, by itself, 
twenty per cent larger than all of the Earth’s land area. 
It’s a big glob of water. 

I was aware of this when I crossed the Pacific (well, 
half of it anyway) and I was also aware that when I was 
looking at all that water, I was seeing only the top of it. 

The Pacific Ocean is not only the most spread out of 
the oceans but the deepest, with an average depth of about 
2.6 miles. In comparison, the Indian Ocean has an average 
depth of about 2.4 miles and the Atlantic only about 2.1 
miles. We can therefore work out the volume of the 
different oceans, as in Table 5. 

As you see then the water of the World Ocean is dis- 
tributed among the three oceans in just about the ratio 
of 2:1:1. 

The total, 339,000,000 cubic miles, is a considerable 
amount. It makes up Vsno of the volume of the Earth — a 
most respectable fraction. If it were all accumulated into 
one place it would form a sphere about 864 miles in 
diameter. This is larger than any asteroid in the solar 
system, probably larger than all the asteroids put to- 
gether. 



238 


NUMBERS AND THE EARTH 


Water , Water , — 


239 


Table 5 Fo/wme of the Oceans 


Pacific 

Atlantic 

Indian 

Total 


VOLUME 

(cubic miles) 

177.000. 000 

87.000. 000 

75.000. 000 

339.000. 000 


PER CENT OF 
WORLD OCEAN 

52.2 

25.7 

22.1 


There is therefore no shortage of water. If the oceans 
were divided up among the population of the Earth, 
each man, woman, and child would get a tenth of a cubic 
mile of ocean water. If you think that’s not much (just a 
miserable tenth of a single cubic mile), consider that that 
equals 110,000,000,000 gallons. 


Of course, the ocean consists of sea water, which has 
limited uses. You can travel over it and swim in it — but 
you can’t (without treatment) drink it, water your lawn 
with it, wash efficiently with it, or use it in industrial 
processes. 

For all such vital operations, you need fresh water, 
and there the ready-made supply is much more limited. 
Ocean water (including a bit of inland salt water) makes 
up about 98.4 per cent of all the water on Earth; and 
fresh water makes up 1.6 per cent or about 5,800,000 
cubic miles. 

That doesn’t sound too bad, but it’s not the whole 
story. Fresh water exists in three phases, solid, liquid, 
and gaseous. (And, incidentally, let me interrupt myself 
to say that water is the only common substance on Earth 
that exists in all three phases; and the only one to exist 
chiefly in the liquid phase. All other common substances 
either exist solely in the gaseous state, as do oxygen and 
nitrogen; or solely in the solid state, as do silica and 
hematite.) The distribution of the fresh water supply of 
the Earth among the three phases is as shown in Table 6: 

Table 6 The Fresh Water Supply 

VOLUME 
(CUBIC MILES) 

Ice 5,680,000 

Liquid fresh water 120,000 

Water vapor (if condensed to liquid) 3,400 


Most of Earth’s supply of fresh water is unavailable to 
us because it is tied up as ice. It is, of course, quite pos- 
sible and even simple to melt ice, but the problem is one 
of location. Nearly ninety per cent of the world’s ice is 
compacted into the huge ice cap that covers Antarctica 
and most of the rest into the smaller sheet that covers 
Greenland. What’s left (about 200,000 cubic miles) occurs 
as glaciers in the higher mountains and the smaller Arctic 
islands plus some polar sea ice. All of this ice is quite 
out of the way. 

That leaves us with just under 125,000 cubic miles of 
fresh water in liquid and gaseous form, and this repre- 
sents the most valuable portion of the water resources of 
the planet. The fresh water supply is constantly running 
off into the sea through flowing rivers and seeping ground 
water or evaporating into the air. This loss, however, is 
constantly replaced by rainfall. It is estimated that the 
total rainfall on all the land areas of the world amounts 
to 30,000 cubic miles per year. This means that one- 
quarter of the fresh water supply is replaced each year and 
if there were no rain at all anywhere, Earth’s dry land 
would become dry indeed, for in four years (if one as- 
sumes that the rate of flow, seepage, and evaporation 
remains constant) fresh water would be all gone. 

If Earth’s fresh water were evenly distributed among 
humanity, every man, woman, and child would own 

40,000,000 gallons and every year he could use 10,000,- 
000 gallons Of his supply, collecting rain replacement in 
return. 

But alas, the fresh water is not evenly distributed. 
Some areas on Earth have a far greater supply than they 
can use and other areas are parched. The maldistribution 
works in time as well as in space; for an area which is 
flooded one year may be drought-stricken the next. 

The most spectacular reservoirs of fresh water are the 
lakes of the world. Of course, not all enclosed bodies of 
water are fresh. Only those bodies of water are fresh 
which have outlets to the ocean so that the efflux of water 
removes the salt dissolved out of the land and brought 
into the lake. Where a lake has no outlet to the ocean, it 
can lose water only through evaporation and the dissolved 
salts do not evaporate. More salt is constantly brought in 
by rivers feeding into the enclosed body and the result 



240 


NUMBERS AND THE EARTH 


Water, Water , Everywhere — 


241 


is a salt lake which, in some cases, is far saltier than the 
sea. 

In fact, the largest inland body of water in the world 
— the Caspian Sea, located between the Soviet Union and 
Iran — is not fresh water. It has an area of 169.381 square 
miles, just about the size of California and it has 3,370 
miles of shoreline. 

It is sometimes stated that the Caspian Sea is not a sea 
but merely a lake, although a very large one. However, 
it seems to me that “lake’' might well be restricted to 
enclosed bodies of fresh water. If “sea 1 ’ is taken to mean 
salt water, whether in the ocean or not, then the Caspian 
is indeed the Caspian Sea. 

The Caspian Sea is only 0.6 per cent salt (as compared 
to the 3.5 per cent salt of the oceans) but this is enough 
to make the waters of the Caspian undrinkable, except in 
the northwest corner where the fresh waters of the Volga 
River are discharged. 

About 150 miles east of the Caspian is the Aral Sea, 
which is about LI per cent salt. It is twice as salty as the 
Caspian Sea, but it is much smaller in extent, with a 
surface area of only about 26,000 square miles— though 
that is enough to make it the fourth-largest body of 
enclosed water in the world. 

There are two other notable enclosed bodies of salt 
water. One is the Great Salt Lake (which I would much 
prefer to call the Utah Sea, since it is not “great,” nor, 
by my definition, a lake) and the other is the Dead Sea, 
The Great Salt Lake is only 1,500 square miles in area 
and the Dead Sea is smaller still — 370 square miles. The 
Dead Sea is not much larger, in fact, than the five boroughs 
of New York City. 

Nevertheless, these two relatively small bodies of water 
are unusual for extreme salinity. The Great Salt Lake is 
about fifteen per cent salt and the Dead Sea is about 
twenty-five per cent salt, four times and seven times 
(respectively) as salty as the ocean. 

However, looking at the surface of the water can be 
deceptive. How deep are these four inland seas? From 
data on the depth, we can work out the volume of each 
and the total salt * content, as in Table 7: 


* The salt is not entirely sodium chloride by any means, but that’s an- 
other matter. 


Table 7 The Inland Seas 



AVERAGE 

VOLUME 

TOTAL SALT 


DEPTH (FEET) 

(CUBIC MILES) (TONS) 

Caspian Sea 

675 

21,600 

600,000,000,000 

Aral Sea 

53 

260 

13,000,000,000 

Dead Sea 

1,080 

75 

86,500,000,000 

Great Salt Lake 20 

5.7 

4,000,000,000 


As you see, the tiny Dead Sea isn’t so tiny after all. In 
terms of qualtity of water it is much larger than the 
Great Salt Lake, and it contains 6L> times as much salt 
as the apparently much larger Aral Sea, 

But let’s turn to the true lakes — the enclosed bodies of 
fresh water. The largest such body in terms of surface 
is Lake Superior, which is about as large as the state of 
South Carolina. It is usually listed as the second-largest 
enclosed body of water on Earth (though, as I will show, 
it isn’t really) „ It is, to be sure, a very poor second to the 
mighty Caspian, covering less than one-fifth the area of 
that body of water but remember, the water of Lake 
Superior is fresh. 

Lake Superior is, however, only one of five American 
Great Lakes that are usually treated as separate bodies 
of water but which are neighboring and interconnected 
so that it is really quite fair to consider them all as 
making up one huge basin of fresh water. Statistics con- 
cerning them follow in Table 8: 

Table 8 The American Great Lakes 



AREA 


AVERAGE 

VOLUME 


(SQUARE 

RANK 

DEPTH 

(cubic 


MILES) 

IN SIZE 

(feet) 

miles) 

Superior 

31,820 

2 

900 

5,400 

Huron 

23,010 

5 

480 

2,100 

Michigan 

22,400 

6 

600 

2,600 

Erie 

9.940 

12 

125 

240 

Ontario 

7,540 

14 

540 

770 

Total 

94,710 



11,110 


Taken as a unit, as they should be, the American 
Great Lakes have a little over half the surface area and 



242 


NUMBERS AND THE EARTH 


Water, Water , Everywhere — 


243 


THE DEAD SEA 

The Dead Sea is perhaps the most famous small body of 
water in the world . Although it is mentioned in the Bible 
(after all r Israel, both ancient and modern, borders on it) 
it is never called the Dead Sea . That is the name given it 
later by Greek geographers who were impressed by the fact 
that it contained no life. In the Bible it is called the Salt 
Sea. 

The Jordan River (the most famous small river in the 
world) empties into the Dead Sea, descending into the 
Great Rift Valley, which is gradually splitting eastern 
Africa away from the rest of the continent and will some - 
day form a new ocean of which the Dead Sea is now the 
chief portion. By the time it enters the Dead Sea, the 
Jordan River is 1,286 feet below sea level. The shores of 
the Dead Sea are the lowest regions on Earth. Despite 
that , the greatest depth of the Dead Sea below its water 
level is 1,310 feet . If the Dead Sea and the regions about 
it were filled with water to sea level , the maximum depth 
of water would be 2,600 feet, or just about half a mile . 

The Dead Sea is divided into two unequal parts by a 
small peninsula that extends into it from the eastern shore . 
The northern part , making up about two thirds of the 
whole area, is the deep portion. The southern part , making 
up the remaining third, is quite shallow, with depths of 
from three to thirty feet . 

Some people speculate that the shallow third was 
flooded with water as a result of an earthquake that broke 
the barrier protecting it from the northern portion of the 
lake; that the earthquake may have accompanied a volcanic 
eruption; that there were settlements in the southern por- 
tion when it was dry; that t in short, this accounts for the 
biblical account of the destruction of Sodom and Go- 
morrah. There is no concrete evidence for this, however. 



The Granger Collection 


244 


NUMBERS AND THE EARTH 


volume of the Caspian Sea, And they contain nearly 
one-tenth the total fresh water supply of the planet. 

The only other group of lakes that can even faintly 
compare with the American Great Lakes is a similar 
series, considerably more separated, in East Africa. The 
three largest are Lakes Victoria, Tanganyika, and Nyasa, 
which I can lump together as the African Great Lakes. 
Here are the statistics in Table 9: 

Table 9 The African Great Lakes 



AREA 


AVERAGE 

VOLUME 


(square 

RANK 

DEPTH 

(CUBIC 


miles) 

IN SIZE 

(feet) 

MILES) 

Victoria 

26,200 

3 

240 

1,200 

Tanganyika 

12,700 

8 

1,900 

4,500 

Nyasa 

11,000 

10 

1,800 

3,800 


49,900 



9,500 


The African Great Lakes (two of them at least) are 
remarkable for their depth, so that although they occupy 
an area of only slightly more than half that of the Ameri- 
can Great Lakes, the volume of fresh water they contain 
almost rivals that present in our own larger but shallower 
lakes. 

But if we're going to talk about deep lakes, we've got 
to mention Lake Baikal in south-central Siberia. Its area 
is 13,197 square miles, making it the seventh-largest body 
of enclosed water on Earth, by the usual criterion of 
surface area. Its average depth, however, is 2,300, making 
in the deepest lake in the world. (Its maximum depth is 
4,982 feet or nearly a mile. It is so deep, I was once told, 
that it is the only lake which contains the equivalent of 
deep-sea fish. If so, these are the only fresh-water, deep- 
sea fish in the world.) 

Its depth means that Baikal contains 5,750 cubic miles 
of fresh water, more than that in Lake Superior. 

The only remaining lakes that would fall in the category 
of "great lakes” are three in western Canada. The statistics 
on the average depth of these Canadian Great Lakes are 
virtually nonexistent. I have the figures on maximum 
depth for two of them and nothing at all for the third. 
However, I shall make what I hope is an intelligent guess 


Water , Water , Everywhere — 245 

just to see what things look like, and you will find that 
in Table 10. 

Table 10 The Canadian Great Lakes 



AREA 


AVERAGE 

VOLUME 


(square 

RANK 

DEPTH 

(cubic 


miles) 

IN SIZE 

(feet) 

MILES) 

Great Bear 

12,200 

9 

240 

525 

Great Slave 

10,719 

11 

240 

510 

Winnipeg 

9,460 

13 

50 

90 


Now we are in a position to list the bodies of enclosed 
water in the order of their real size, their fluid contents 
rather than their surface area. To be sure, surface area of 
any lake can be determined with reasonable accuracy, 
whereas the fluid contents can only be roughly estimated, 
so it makes sense to list them in order of decreasing 
surface area. However, I will do as I choose. The four- 
teen largest bodies of enclosed water (in terms of surface 
area) rank in terms of fluid content as shown in Table 11. 

Table 11 The Large Lakes of the Earth 

VOLUME 
(CUBIC MILES) 


Caspian 

21,600 

Baikal 

5,750 

Superior 

5,400 

Tanganyika 

4,500 

Nyasa 

3.800 

Michigan 

2,600 

Huron 

2,100 

Victoria 

1,200 

Ontario 

770 

Great Bear 

525 

Great Slave 

510 

Aral 

260 

Erie 

240 

Winnipeg 

90 


This list is not only a rough one, with several of the 
tfo not contain fresh water 



246 


NUMBERS AND THE EARTH 


figures so rough as to be worthless, but in addition there 
are lakes with smaller areas than any on the list which 
are deep enough to deserve a listing somewhere ahead 
of Winnipeg. These include Lake Ladoga and Lake Onega 
in the northwestern stretch of European Russia, and Lake 
Titicaca in the Andes between Bolivia and Peru. 

But what’s the use? All this talk about water isn’t help- 
ing the parching Northeast at all. Indeed, the water level 
in the American Great Lakes has been falling disturbingly 
in recent years, I understand, and even the Caspian Sea 
is shrinking. 

Maybe old Mother Earth is getting tired of us. . . . In 
my more morose moments, I wonder if I could bring 
myself to blame her if she were. 



UP AND 
DOWN 
THE EARTH 


Boston is getting its face lifted, and we now have 
‘The New Boston.’ ’ * 

The outstanding feature of the New Boston is the 
Prudential Center, which is an area of the Back Bay that 
has been renovated into New York-like luxury. It pos- 
sesses a new hotel, the Sherat on-Boston, and, most spec- 
tacular of all, a beautiful skyscraper, the fifty- two-story 
Prudential Tower, which is 750 feet tall. 

In the summer of 1965, I invaded the center for the 
first time. I was asked to join a panel discussion dealing 
with the future of industrial management. The panel was 
held in the Sheraton-Boston under conditions of great 
splendor, and after the dinner that followed, the manager 
of the hotel announced, in the course of a short talk, that 
the Prudential Tower was the tallest office building in 
continental North America. 

We registered amazement and he at once explained 
yes, there were indeed taller office buildings not far 
from Boston, but they were not on continental North 
America. They were on an island off the shores of the 
continent; an island named Manhattan. 

art * cle f irs * a Ppea?ed in February 1966 1 at which time I lived in 

ther*°r*L n ar€a ' In i970, however > 1 moved to New York City and it is 
nere 1 to™ been ever since.] 


247 



248 


NUMBERS AND THE EARTH 


Up and Down the Earth 


249 


And he was right. Outside the island of Manhattan, 
there is, at the moment of writing, no office building 
taller than the Prudential Tower anywhere in North 
America. (Perhaps anywhere in the world.) * 

It made me think at once that you can play a large 
number of games, if you are the record-gathering type 
(as I am), by altering qualifications slightly. Long before 
the manager's speech was over, I was thinking of moun- 
tains. 

Everyone knows the name of the highest mountain of 
the world. It is Mount Everest, located in the Himalayan 
Mountain Range, exactly on the border between Nepal 
and Tibet. 

It is named for a British military engineer, George 
Everest, who spent much of his adult life surveying Java 
and India and who, from 1830 to 1843, was surveyor 
general of India. In 1852, when a mountain was dis- 
covered in the north, one which was at once suspected 
of being height champion, it was named for him. At that, 
his name is easier to pronounce than the native Tibetan 
name for the mountain — Chomolungma. 

The height of Mount Everest is usually given in the 
reference books as 29,002 feet above sea level, a value 
first obtained in 1860, though I believe that the most 
recent trigonometric measurements make it 29,141 feet. 
In either case, the tippy, tippy top of Mount Everest is 
the only piece of solid land on the face of the globe that 
is more than 29,000 feet above sea level, so that the 
mountain qualifies nicely as something quite unique. 
Using another unit of measure. Mount Everest is just a 
trifle more than five and a half miles high and all other 
land is less than five and a half miles above sea level. 

Except by members of the “Anglo-Saxon nations/' 
however, mountain heights are generally measured in 
meters rather than in feet or miles. ** There are 3.28 feet 
in a meter and Mount Everest stands 8,886 meters above 
sea level. 

At once this gives rise to the question: how many 
other mountains are there that belong to the rarefied 


l* Since this article appeared, two office buildings higher than ihe Pru- 
dential Tower have been constructed in Chicago .] 

[** Nowadays, even the Anglo-Saxon nations are falling into line . Right 
now t the United States is the only holdout of consequence .] 


aristocracy of those that tower more than 8,000 meters 
above sea level. The answer is; Not many. Just thirteen! * 
/md here they are in Table 12. 

Of these thirteen aristocrats all but four are in the 
Himalaya Mountain Range, spread out over a stretch of 
a little over three hundred miles. The tallest exception is 
Mount Godwin Austen, which is named for Henry Haver- 
sham Godwin-Austen, another Britisher who was engaged 
in the nineteenth-century trigonometric surveys of India. 
It is only recently that the mountain came to be officially 
known by his name. Previously, it was known simply as 
K-2> Its native name is Dapsang, 

Mount Godwin Austen is located about eight hundred 
miles northwest of Mount Everest and the other Hima- 
layan towers. It is the highest peak of the Karakorum 
Mountain Range, running between Kashmir and Sinkiang. 

Table 12 The 8,000-Meter Mountains 


MOUNTAINS 

FEET 

HEIGHT 

MILES 

METERS 

Everest 

29,141 

5.52 

8,886 

Godwin Austen 

28,250 

5.36 

8,613 

Kanchenjunga 

28,108 

5.33 

8,570 

Lhotse 

27,923 

5.29 

8,542 

Makalu 

27,824 

5.28 

8,510 

Dhaulagiri 

26,810 

5.10 

8,175 

Manaslu 

26,760 

5.06 

8,159 

Cho Oyu 

26,750 

5.06 

8,155 

Nanga Parbat 

26,660 

5.05 

8,125 

Annapurna 

26,504 

5.03 

8,080 

Gasherbrum 

26,470 

5.02 

8,075 

Broad 

26,400 

5.00 

8,052 

Gosainthan 

26,291 

4.98 

8.016 


All thirteen of the eight -thousanders are located in Asia 
and all are located in the borderlands that separate India 
and China. 

This is true, indeed, not only for the thirteen highest 

[* / m not as confident about that statement now as I was at the time 
the article was written . The Guinness Book of World Records speaks of 
Casherbrum as the fifteenth tallest mountain and if that is so (and I trust 
uinness) then there are at least four 8,000-meter mountains I have not 
listed m the table and which I cannot find in my library .] 



250 


NUMBERS AND THE EARTH 


Up and Down the Earth 


251 


but for the sixty highest (!) mountains in the world, at 
least, so that the area is the place for mountaineers.* 

And of all mountains, Everest is obviously the moun- 
tain to climb. The first serious attempt to climb it was 
made in 1922 and after a full generation of effort, eleven 
lives had been lost on its slopes and no successes were 
scored. Then, on May 29, 1953, the New Zealander Ed- 
mund Percival Hillary and the Sherpa Tenzing Norgay 
made it. Since then, others have too. 

You would think that with even Everest conquered, 
there would remain no more mountains unclimbed, but 
that is not so. Everest received a lot more attention than 
some of the other peaks. As of now (unless someone has 
sneaked up the slopes while I wasn’t looking) the highest 
mountain still unconquered is Gosainthan, which is only 
the thirteenth highest.** 

The highest mountain range outside Asia is the Andes 
Mountain Range, running down the western edge of South 
America. The highest peak in the Andes is Mount Acon- 
cagua, which stands 22,834 feet high. Despite the fact 
that Mount Aconcagua is the highest mountain peak in 
the world outside Asia, there are scores of higher peaks 
in Asia. 

Table 13 The Highest Mountains by Regions 


REGION 

MOUNTAIN 

FEET 

HEIGHT 

MILES 

METERS 

Asia 

Everest 

29,141 

5.52 

8,886 

South America 

Aconcagua 

22,834 

4.34 

6,962 

North America 

McKinley 

20,320 

3.85 

6,195 

Africa 

Kilimanjaro 

19,319 

3.67 

5,890 

Europe 

Elbrus 

18,481 

3.50 

5,634 

Antarctica 

Vinson Massif 

16,860 

3.19 

5,080 

48 States 

Whitney 

14,496 

2.75 

4,419 

Australia 

Kosciusko 

7,328 

1.39 

2,204 

New England 

Washington 

6,288 

1.19 

1,918 


[* The Himalayas, I have recently been told , contain 96 of the JOS highest 
peaks in the world,] 

\ ** The Guinness Book of World Records says it is Gasherbrum, the 
fifteenth highest.] 


For the records, here, in Table 13, are the highest 
peaks in each of the continents. To soothe my national 
and regional pride, I will add the highest mountain in the 
forty-eight contiguous states, and in New England, too. 
(After all, I'm writing the chapter and can do as I please.) 

To locate these mountains — 

Mount Aconcagua is in Argentina, very close to the 
border of Chile, only a hundred miles east of Valparaiso. 

Mount McKinley is in south-central Alaska, about one 
hundred fifty miles southwest of Fairbanks. The fact that 
it was the highest land in North America was discovered 
in 1896 and it was named after William McKinley, who 
had just been elected President of the United States. The 
Russians (who had owned Alaska before 1867) had called 
it “Bolshaya” (“large”). 

Mount Kilimanjaro is in northeastern Tanganyika, near 
the border of Kenya, and is about two hundred miles 
from the Indian Ocean. Mount Elbrus is in the Caucasus 
Mountain Range, about sixty miles northeast of the Black 
Sea. 

Concerning the Vinson Massif, alas, I know virtually 
nothing. Even its height is only a rough estimate. 

Mount Whitney is in California, on the eastern border 
of Sequoia National Park. It is only eighty miles west of 
Death Valley in which is to be found the lowest point of 
land in the forty-eight states (a pool called Badwater — 
and I’ll bet it is — which is two hundred eighty feet below 
sea level). Mount Whitney is named for the American 
geologist Josiah Dwight Whitney, who measured its height 
in 1864. 

Mount Kosciusko is in the southeastern corner of Aus- 
tralia, on the boundary between the states of Victoria and 
New South Wales? It is the highest point of the range 
called the Australian Alps. I suspect it was discovered at 
the end of the eighteenth century, when the Polish patriot 
Thaddeus Kosciusko was leading the last, forlorn fight for 
Polish independence, but I can’t be sure. 

Mount Washington is located in the Presidential Moun- 
tain Range in northern New Hampshire, and we all know 
who it is named after. 

By listing the high mountains by continents, I don’t 
jnean to imply that all high mountains are on continents. 
In fact, Australia, usually considered a continent (though 
* small one) possesses no particularly high mountains, 



252 


NUMBERS AND THE EARTH 


Up and Down the Earth 


253 


while New Guinea to its north (definitely an island, 
though a large one) is much more mountainous and 
possesses dozens of peaks higher than any in continental 
Australia, and some that are quite respectable by any 
standards. Three Pacific islands are notably mountainous 
and these are listed in Table 14. 

Table 14 Notable Island Mountains 


ISLAND 

MOUNTAIN 

FEET 

HEIGHT 

MILES 

METERS 

New Guinea 

Carstensz 

16,404 

3.12 

5,000 

Hawaii 

f Mauna Kea 

13,784 

2.61 

4,200 

\ Mauna Loa 

13.680 

2.59 

4,171 

Sumatra 

Kerintji 

12,484 

2.36 

3,807 

New Zealand 

Cook 

12.349 

2.34 

3,662 


Mount Carstensz is the highest mountain in the world 
that is not on a continent. Whom it is named for I do not 
know, but it is in the western portion of New Guinea 
and is part of the Nassau Mountain Range, which is named 
for the Dutch royal family. I suspect that by now Indo- 
nesia has renamed both the range and the mountain or, 
more likely, has restored the original names, but I don’t 
know what these might be.* 

Mount Cook is just a little west of center in New 
Zealand's southern island. It is named for the famous 
explorer Captain Cook, of course, and its Maori name 
is Aorangi. 

All the heights I have given for the mountains, so far. 
are “above sea level.” 

However, remembering the manager of the Sheraton- 
Boston Hotel, let’s improve the fun by qualifying matters. 

After all. the height of a mountain depends a good deal 
upon the height of its base. The Himalayan mountain 
peaks are by far the most majestic in the world; there is 
no disputing that. Nevertheless, it is also true that they 
sit upon the Tibetan plateau, which is the highest in the 
world. The Tibetan “lowlands” are nowhere lower than 
some 12,000 feet above sea level. 


[* I have found out since. Mount Carstensz is now officially Mount 
Djaja of the Sudlrman Range,'] 


If we subtract 12,000 feet from Mount Everest’s height, 
we can say that its peak is only 17,000 feet above the 
land mass upon which it rests. 

This is not exactly contemptible, but by this new 
standard (base to top, instead of sea level to top) are 
there any mountains that are higher than Mount Everest? 
Yes, indeed, there is one, and the new champion is not 
in the Himalayas, or in Asia, or on any continent. 

This stands to reason after all. Suppose you had a 
mountain on a relatively small island. That island may be 
the mountain, and the mountain wouldn’t look impressive 
because it was standing with its base in the ocean depth 
and with the ocean lapping who knows how many feet 
up its slopes. 

This is actually the case for a particular island. That 
island is Hawaii — the largest single unit of the Hawaiian 
Islands. The island of Hawaii, with an area of 4,021 
square miles (about twice the size of Delaware) is actually 
a huge mountain rising out of the Pacific. It comes to 
four peaks, of which the two highest are Mauna Kea 
and Mauna Loa ( See Table 13). 

The mountain that makes up Hawaii is a volcano ac- 
tually, but most of it is extinct. Mauna Loa alone remains 
active. It, all by itself, is the largest single mountain in 
the world in terms of cubic content of rock, so you can 
imagine how large the whole mountain above and below 
sea level must be. 

The central crater pit of Mauna Loa is sometimes 
active but has not actually erupted in historic times. In- 
stead, the lava flow comes from openings on the sides. 
The largest of these is Kilauea, which is on the eastern 
side of Mauna Loa, some 4,088 feet (0.77 mile, or 
1,246 meters) above sea level. Kilauea is the largest 
active crater in the world and is more than two miles 
in diameter. 

As though these distinctions are not enough, this tre- 
mendous four-peaked mount we call Hawaii becomes 
totally astounding if viewed as a whole. If one plumbs the 
ocean depths, one find that Hawaii stands on a land base 
that is over 18,000 feet below sea level. 

If the oceans were removed from Earth’s surface (only 
temporarily, please), then no single mountain on Earth 
could possibly compare with the breathtaking towering 
majesty of Hawaii. It would be by far the tallest moun- 



254 


Up and Down the Earth 


255 


numbers and the earth 

tain on Earth, counting from base to peak. Its height on 
that basis would be 32,036 feet (6.08 miles or 9,767 
meters). It is the only mountain on earth that extends 
more than six miles from base to tip. 

The vanishing of the ocean would reveal a similar, 
though smaller, peak in the Atlantic Ocean; one that is 
part of the Mid-Atlantic Mountain Range. For the most 
part, we are unaware of the mountain range because it 
is drowned by the ocean, but it is larger and longer and 
more spectacular than any of the mountain ranges on 
dry land, even than the Himalayas. It is 7,000 miles long 
and 500 miles wide and that’s not bad. 

Some of the highest peaks of the range do manageto 
poke their heads above the surface of the Atlantic. The 
Azores, a group of nine islands and several inlets (be- 
longing to Portugal), are formed in this manner. They 
are about 800 miles west of Portugal and have a total 
land area of 888 square miles, rather less than that of 

Rhode Island. , , . , , . . 

On Pico Island in the Azores stands the highest point 
of land on the island group. This is Pico Alto ( High 
Mountain”), which reaches 7,460 feet (1.42 miles or 
2,274 meters) above sea level. However, if you slide down 
the slopes of the mountain and proceed all the way down 
to the sea bottom, you find that only one-quarter of the 
mountain shows above water. 

The total height of Pico Alto from underwater base to 
peak is about 27,500 feet (5.22 miles or 8,384 meters), 
which makes it a peak of Himalayan dimensions. 

While we have the oceans temporarily gone from the 
Earth, we might as well see how deep the ocean goes 
About 1.2 per cent of the sea bottom lies more than 
6 000 meters below sea level, and where this happens we 
have the various "Trenches ” There are a number of these, 
most of them in the Pacific Ocean. All are near island 
chains and presumably the same process that burrows 
out the deeps also heaves up the island chain. 

The greatest depth so far recorded in some of these 
deeps (according to the material available to me) is 

given in Table 15. 

The figures for the depth of the deeps are by no means 
as reliable as those for the heights of mountains, of course, 
and I can’t tell when some oceanographic ship will plumb 


a deeper depth in one or more of these deeps. The 
greatest recorded depth in the Mindanao Trench — and in 
the world — was plumbed only as recently as March, 1959, 
by the Russian oceanographic vessel the Vityaz . 

The greatest depth in the Mariana Trench was actually 
reached by Jacques Piccard and Don Walsh in person on 
January 23, 1960, in the bathyscaphe Trieste . This has 
been named the “Challenger Deep” in honor of the ocean- 
ographic vessel the H.M.S. Challenger f which conducted 
a scientific cruise from 1872 to 1876 all over the oceans 
and established modem oceanography. 

In any case, the oceans are deeper than any mountain 
is high, which is the point I want to make, and in several 
places. 

Consider the greatest depth of the Mindanao Trench. 
If Mount Everest could be placed in it and made to nestle 
all the way in, the mountain would sink below the waves 
and waters would roll for 7,000 feet ( 1 % miles) over its 
peak. If the island of Hawaii were moved from its present 
location, 4,500 miles westward, and sunk into the Min- 
danao Trench, it too would disappear entirely and 4,162 
feet of water (% of a mile) would flow above its tip. 

Table 15 Some Ocean Trenches 



GENERAL 

DEPTH 


TRENCH 

LOCATION 

FEET 

MILES 

METERS 

Bartlett 

S. of Cuba 

22,788 

4.31 

6,948 

Java 

S. of Java 

24,442 

4.64 

7,252 

Puerto Rico 

N. of Puerto Rico 

30,184 

5.71 

9,392 

Japan 

S. of Japan 

32.153 

6.09 

9,800 

Kurile 

E, of Kamchatka 

34,580 

6.56 

10,543 

Tonga 

E. of New Zealand 

35,597 

6.75 

10,853 

Mariana 

E. of Guam 

35,800 

6.79 

10,915 

Mindanao 

E. of Philippines 

36,198 

6.86 

11,036 


If sea level is the standard, then the lowest bit of the 
solid land surface on Earth, just off the Philippines, is 
about 3,200 miles east of the highest bit at the top of 
Mount Everest. The total difference in height is 65,339 
feet (12.3 miles or 19,921 meters). 

This sounds like a lot but the diameter of the Earth 
is about 7,900 miles so that this difference in low-high 
makes up only 0,15 percent of the Earth’s total thickness. 



256 


NUMBERS AND THE EARTH 


KILAUEA 


Manna Loa is the largest mountain mass in the world and 
together with Mauna Kea virtually makes up the island 
of Hawaii. Mauna Loa means '‘long mountain” which is 
a good name since from end to end it is seventy-five miles 
wide, Mauna Kea means "white mountain" because it is 
usually snowcapped. The snowcap indicates that Mauna 
Kea is a dormant volcano, but Mauna Loa is active , and 
near it is Kilauea ( here illustrated}, which is the largest 
crater in the world , with an area of four square mites. 

Kilauea is not known for spectacular eruptions. Rather, 
it is always simmering , ar.d within the crater is a boning 
lake of molten rock that occasionally rises and overflows 
its sides . Occasionally such a lava flow is copious and 
long-sustained and in 1935 even threatened the city of 
Hilo , largest on the island. 

The native Hawaiians considered the most active vent 
of Kilauea to be the home of Pele, the fire goddess 
which makes sense if we assume the existence of divine 

beings . . , . 

Here t however , rests a peculiar coincidence . I he 
Hawaiian fire goddess is Pele , and on the island of Mar- 
tinique is a mountain called Pelee . The two names have 
nothing in common despite appearances, since 'Montague 
Pelee” is merely “ Bald Mountain ” in French (or "peeled 
mountain, if you want to preserve the sound), because the 

top was denuded. , , 

The name, however , fateful, for the mountain is a 

volcano and that is why the top is denuded . It was not 
taken as a very serious volcano , for two minor eruptions t 
one in 1792 and one in 1851, impressed nobody. But them 
on May 8 , 1902 , the mountain suddenly exploded . An 
avalanche of lava went rolling down the side right down 
onto St. Pierre f then the capital of Martinique. One mo- 
ment it was a city of 30,000 people; the next a collection 
of 30,000 corpses. Only two people survived . 



258 


NUMBERS AND THE EARTH 


If the Earth were shrunk to the size of my library 
globe (16 inches in diameter), the peak of Mount Everest 
would project only 0.01 inch above the surface and the 
Mindanao Trench would sink only 0.014 inch below 
the surface. 

You can see then that despite all the extreme ups and 
downs I have been talking about, the surface of the Earth, 
viewed in proportion to the size of the Earth, is very 
smooth. It would be smooth, even if the oceans were 
gone and the unevenness of the ocean bottom were ex- 
posed. With the oceans filling up most of the Earth’s 
hollows (and concealing the worst of the unevennesses), 
what remains is nothing. 

But let’s think about sea level again. If the Earth con- 
sisted of a universal ocean, it would take on the shape 
of an ellipsoid of revolution, thanks to the fact that the 
planet is rotating. It wouldn’t be a perfect ellipsoid be- 
cause, for various reasons, there are deviations of a few 
feet here and there. Such deviations are, however, of only 
the most academic interest and for our purposes we can 
be satisfied with the ellipsoid. 

This means that if Earth were bisected by a plane cut- 
ting through the center, and through both poles, the cross- 
sectional outline would be an ellipse. The minor axis 
(or shortest possible Earth-radius) would be from the 
center to either pole, and that would be 6,356,912 meters. 
The longest radius, or major axis, is from the center to 
any point on the equator. This is 6,378,388 meters (on 
the average, if we wish to allow for the fact that tne 
equator is itself very slightly elliptical). 

The equatorial sea level surface, then, is 21,476 meters 
(70 000 feet or 13.3 miles) farther from the center than 
the polar sea level surface is. This is the well-known 
“equatorial bulge.” 

However, the bulge does not exist at the equator alone. 
The distance from center to sea level surface increases 
smoothly as one goes from the poles to the equator. Un- 
fortunately, I have never seen any data on the extra 
length of the radius (over and above its minimum lengtn 
at the poles) for different latitudes. 

I have therefore had to calculate it for myself, making 
use of the manner in which the gravitational field vanes 
from latitude to latitude. (I could find figures on tne 


Up and Down the Earth 


259 


gravitational field.) The results, which I hope are approxi- 
mately right anyway, are included in Table 16. 

Table 16 The Earth 9 s Bulge 


EXTRA LENGTH OF EARTH'S RADIUS 


LATITUDE 

FEET 

MILES 

METERS 

0° (equator) 

70,000 

13.3 

21,400 

5° 

69,500 

13.2 

21,200 

10° 

68,000 

12.9 

20,800 

15° 

65,500 

12.4 

20,000 

20° 

62,300 

11.8 

19,000 

25° 

58,000 

11.0 

17,700 

30° 

52,800 

10.0 

16,100 

35° 

47,500 

9.0 

14,500 

40° 

41,100 

7.8 

12,550 

45° 

35,100 

6.65 

10,700 

50° 

29,000 

5.50 

8,850 

55° 

23,200 

4.40 

7,050 

60° 

17,700 

3.35 

5,400 

65° 

12,500 

2.37 

3,800 

70° 

8,250 

1.56 

2,500 

75° 

4,800 

0.91 

1,460 

80° 

2,160 

0.41 

660 

85° 

530 

0.10 

160 

90° (poles) 

0 

0.00 

0 


Suppose, now, we measure heights of mountains not 
from just any old sea level, but from the polar sea level. 
This would serve to compare distances from the center 
of the Earth , and certainly that is another legitimate way 
of comparing mountain heights. 

If we did this, we would instantly get a completely 
new perspective on matters. 


mv lumuajiau Uipa UUWli LU I i ,UJ O 

meters below sea level; but that means below the sea 
level at its own latitude, which is 10.0° N. That sea level 
is 20,800 meters above the polar sea level so that the 
greatest depth of the Mindanao Trench is still some 9,800 
meters (6.1 miles) above the polar sea level. 

Nr &r words> wllen p eary stood on the sea ice at the 
Farfi? e> was m ^ es closer to the center of the 
th» k ^ bad been in a bathyscaphe probing 
bottom of the Mindanao Trench. 



260 


NUMBERS AND THE EARTH 


Up and Down the Earth 


261 


Of course, the Arctic Ocean has a depth of its own. 
Depths of 4,500 meters (2.8 miles) have, I believe, been 
recorded in the Arctic. This means that the bottom of the 
Arctic Ocean is nearly nine miles closer to the center of 
the Earth than the bottom of the Mindanao Trench, and 
from this point of view, we have a new candidate for the 
mark of “deepest deep.” (The south polar regions are 
filled up by the continent of Antarctica, so it is out of 
the running in this respect.) 

And the mountains? 

Mount Everest is at a latitude of about 30.0 a . Sea level 
there is 16,100 meters higher than polar sea level. Add 
that to the 8,886 meters that Mount Everest is above its 
own sea level mark and you find that the mountain is just 
about 25,000 meters (15.5 miles) above polar sea level. 
But it is only 2.2 miles above equatorial sea level. 

In other words, when a ship is crossing the equator, 
its passengers are only 2.2 miles closer to the center of 
the Earth than Hillary was when he stood on Mount 
Everest’s peak. 

Are there mountains that can do better than Mount 
Everest by this new standard? The other towers of Asia 
are in approximately Mount Everest’s latitude. So are 
Mount Aconcagua and some of the other high peaks of 
the Andes (though on the other side of the equator). 

Mount McKinley is a little over 60.0" N so that its sea 
level is only some 5,000 meters above polar sea level. Its 
total height above polar sea level is only 11,200 meters 
(7.0 miles), which is less than half the height of Mount 
Everest. 

No, what we need are some good high mountains near 
the equator, where they can take full advantage of the 
maximum bulge of the Earth’s midriff, A good candidate 
is the tallest mountain in Africa, Mount Kilimanjaro. It 
is about 3.0° S and is 5,890 meters high. To this one can 
add the 21,300-meter-high bulge it stands on, so that it 
is some 27,200 meters above polar sea level (16.9 miles), 
or nearly a mile and a half higher than Mount Everest, 
counting from the center of the Earth. 

And that is not the best, either. My candidate for 
highest peak by these standards is Mount Chimborazo 
in Ecuador. It is part of the Andes Mountain Range, in 
which there are at least thirty peaks higher than Mount 


Chimborazo. Mount Chimborazo, is, however, at 2.0° S. 
Its height above its own sea level is 6,300 meters. If we 
add the equatorial bulge, we have a total height of 27,600 
meters above polar s^a level (17.2 miles).* 

If we go by the distance from the center of the Earth, 
then, we can pass from the bottom of the Arctic Ocean 
to the top of Mount Chimborazo and increase that dis- 
tance by 32,100 meters, or just about 20.0 miles— a nice 
even number. 

By changing the point of view then, we have three 
different candidates for tallest mountain on Earth: 
Mount Everest, Maun a Kea, and Mount Chimborazo. We 
also have two different candidates for the deepest deep: 
the bottom of the Arctic Ocean and the bottom of the 
Mindanao Trench. 

But let’s face it. What counts in penetrating extreme 
depths or extreme heights is not mere distance, but diffi- 
culty of attainment. The greatest single measure of diffi- 
culty in plumbing depths is the increase of water pressure; 
and the greatest single measure of difficulty in climbing 
heights is the decrease of air pressure. 

By that token, water pressure is highest at the bottom 
of the Mindanao Trench, and air pressure is lowest at 
the top of Mount Everest, and therefore those are the 
extremes in practice. 


[* Since this article first appeared Chimborazo has been announced as 
the highest mountain in the world by others and the news has been 
printed with considerable exciiement by Scientific American and by News' 
week. Neither took note of this article's precedence ,] 



The Isles of Earth 


263 


THE 

ISLES 

OF EARTH 


One of the nicest things about the science essays 
I write is the mail it brings me — almost invariably good- 
humored and interesting, 

Consider, for instance, the previous chapter “Up and 
Down the Earth,' 1 in which I maintain that Boston's 
Prudential is the tallest office building on continental 
North America (as opposed to the higher ones on the 
island of Manhattan). The moment that essay first ap- 
peared I received a card from a resident of Greater 
Boston, advising me to follow the Charles and Neponset 
rivers back to their source and sec if Boston could not 
be considered an island. 

I followed his advice and, in a way, he was right. The 
Charles River flows north of Boston and the Neponset 
River flows south of it. In southwestern Boston they ap- 
proach within two and one-half miles of each other. 
Across that gap there meanders a stream from one to 
the other so that most of Boston and parts of some 
western suburbs (including the one I live in)* are sur- 
rounded on all sides by surface water. The Prudential 
Tower, and my house, too, might therefore be considered, 
by a purist, to be located on an island. 


[* Not anymore. As l said before, I’ve returned to New York.] 


Well! 

But before I grow panicky, let me stop and consider. 
What is an island, anyway? 

The word “island” comes from the Anglo-Saxon 
“eglond,” and this may mean, literally, “water-land”; that 
is “land surrounded by water.” 

This Anglo-Saxon word, undergoing natural changes 
with time, ought to have come down to us as “eyland” or 
“Hand.” An “s” was mistakenly inserted, however, though 
the influence of the word “isle,” which is synonymous 
with “island” yet, oddly enough, is not etymologically 
related to it. 

For “isle” we have to go back to classical times. 

The ancient Greeks in their period of greatness were a 
seafaring people who inhabited many islands in the 
Mediterranean Sea as well as sections of the mainland. 
They, and the Romans who followed, were well aware 
of the apparently fundamental difference between the two 
types of land. An island was to them a relatively small 
bit of land surrounded by the sea. The mainland (of 
which Greece and Italy were part) was, on the other 
hand, continuous land with no known boundary. 

To be sure, the Greek geographers assumed that the 
land surface was finite and that the mainland was sur- 
rounded on all sides by a rim of ocean but, except for 
the west, that was pure theory. In the west, beyond the 
Strait of Gibraltar, the Mediterranean Sea did indeed 
open up into the broad ocean. No Greek or Roman, how- 
ever, succeeded in traveling overland to Lapland, South 
Africa, or China, as to stand on the edge of land and, 
with his own eyes, gaze at the sea. 

In Latin, then, the mainland was terra continens; that 
is, “land that holds together,” The notion was that when 
you traveled on the mainland, there was always another 
piece of land holding on to the part you were traversing. 
There was no end. The phrase has come down to us as 
the word “continent.” 

On the other hand, a small bit of land, which did not 
hold together with the mainland, but which was separate 
and surrounded by the sea was terra in salo or “land in 
the sea.” This shortened to insula in Latin; and, by suc- 
cessive steps, to isola in Italian, “isle” in English, and 
ile in French. 



262 



264 


NUMBERS AND THE EARTH 


The strict meaning of the word “isle,” then (and by 
extension the word “island”), is that of land surrounded 
by salt water. Of course, this is undoubtedly too strict. 
It would render Manhattan's status rather dubious since 
it is bounded on the west by the Hudson River. Then, 
too, there are certainly bodies of land, usually called 
islands, nestled within lakes or rivers, which are certainly 
surrounded by fresh water. However, even such islands 
must be surrounded by a thickness of water that is fairly 
large compared to the island’s diameter. No one w r ould 
dream of calling a large tract of land an island just be- 
cause a creek marked it off. So Boston is not an island, 
practically speaking, and Manhattan is. 

However, for the purposes of the remainder of this 
article I am going to stick to the strict definition of the 
term and discuss only those islands that are surrounded 
by salt water — 

If we do this, however, then, again strictly speaking, 
the land surface of the Earth consists of nothing but 
islands. There are no continents in the literal meaning of 
the word. The mainland is never endless. The Venetian 
traveler Marco Polo reached the eastern edge of the an- 
ciently known mainland in 1275; the Portuguese navigator 
Bartholomew Diaz reached the southern edge in 1488; 
and Russian explorers marked off the northern edge in 
the seventeenth and early eighteenth centuries. 

The mainland I refer to here is usually considered as 
making up the three continents of Asia, Europe, and 
Africa. But why three continents, where there is only a 
single continuous sheet of land, if one ignores rivers and 
the man-made Suez Canal? 

The multiplicity of continents dates back to Greek days. 
The Greeks of Homeric times were concentrated on the 
mainland of Greece and faced a hostile second mainland 
to the east of the Aegean. The earliest Greeks had no 
reason to suspect that there was any land connection 
between the two mainlands and they gave them two 
different names; their own was Europe, the other Asia. 

These terms are of uncertain origin but the theory I 
like best suggests they stem from the Semitic words assu 
and erev, meaning “east” and “west” respectively. (The 
Greeks may have picked up these words from the Phoeni- 
cians, by way of Crete, just as they picked up the 
Phoenician alphabet.) The Trojan War of 1200 b.c. begins 


The Isles of Earth 


265 


the confrontation of, literally, the West and the East; a 
confrontation that is still with us. 

Of course, Greek explorers must have learned, early in 
the game, that there was indeed a land connection be- 
tween the two mainlands. The myth of Jason and the 
Argonauts, and their pursuit of the Golden Fleece, prob- 
ably reflects trading expeditions antedating the Trojan 
War. The Argonauts reached Colchis (usually placed at 
the eastern extremity of the Black Sea) and there the two 
mainlands merged. 

Indeed, as we now know, there is some fifteen hundred 
miles of land north of the Black Sea, and a traveler can 
pass from one side of the Aegean Sea to the other — from 
Europe to Asia and back — by way of this fifteen-hund red- 
mile connection. Consequently, Europe and Asia are 
separate continents only by geographic convention and 
there is no real boundary between them at all. The com- 
bined land mass is frequently spoken of as “Eurasia.” 

The Ural Mountains are arbitrarily set as the boundary 
between Europe and Asia in the geography books. Partly, 
this is because the Urals represent a mild break in a 
huge plain stretching for over six thousand miles from 
Germany to the Pacific Ocean, and partly because there 
is political sense in considering Russia (which, until about 
1580, was confined to the region west of the Urals) part 
of Europe. Nevertheless, the Asian portion of Eurasia is 
so much larger than the European portion that Europe 
is often looked upon as a mere peninsula of Eurasia. 

Africa is much more nearly a separate continent than 
Europe is. Its only land connection to Eurasia is the Suez 
Isthmus, which nowadays is about a hundred miles wide, 
and in ancient times was narrower. 

Still the connection was there and it was a well-traveled 
one, with civilized men (and armies too) crisscrossing 
it now and again — whereas they rarely crossed the land 
north of the Black Sea. The Greeks were aware of the 
link between the nations they called Syria and Egypt, and 
they therefore considered Egypt and the land west of it 
to be a portion of Asia. 

The matter was different to the Romans. They were 
farther from the Suez Isthmus and throughout their early 
history that connection was merely of academic interest 
to them. Their connection with Africa was entirely by way 
of the sea. Furthermore, as the Greeks had once faced 



266 


NUMBERS AND THE EARTH 


Troy on opposing mainlands with the sea between, so a 
thousand years later— the Romans faced the Carthaginians 
on opposing mainlands with the sea between. The struggle 
with Hannibal was every bit as momentous to the Romans 
as the struggle with Hector had been to the Greeks. 

The Carthaginians called the region about their city by 
a word which, in Latin, became Africa. The word spread, 
in Roman consciousness, from the immediate neighbor- 
hood of Carthage (what is now northern Tunisia) to the 
entire mainland the Romans felt themselves to be facing. 
The geographers of Roman times, therefore— notably 
the Greek-Egyptian Ptolemy — granted Africa the dignity 
of being a third continent. 

But let’s face facts and ignore the accidents of history. 
If one ignores the Suez Canal, one can travel from the 
Cape of Good Hope to the Bering Strait, or to Portugal 
or Lapland, without crossing salt water, so that the whole 
body of land forms a single continent. This single con- 
tinent has no generally accepted name and to call it 
“Eurafrasia,” as I have sometimes felt the urge to do, 

is ridiculous, , ^ A „ A 

We can think of it this way, though. This tract of land 
is enormous but it is finite and it is bounded on all sides 
by ocean. Therefore it is an island; a vast one, to be sure, 
but an island. If we take that into account then there is 
a name for it; one that is sometimes used by geopoliticians. 

It is "the World Island.” . 

The name seems to imply that the triple continent of 
Europe-A$ia-Africa makes up the whole world and, you 
know, it nearly does. Consider Table 17. (And let me 
point out that in this and the succeeding tables in this 
article, the figures for area are good but those for popu- 
lation are often quite shaky. I have tried to choke mid- 
1960s’ population figures * out of my library, but I 
haven’t always been able to succeed. Furthermore, even 
when such figures are given they are all too frequently 
marked “estimate” and may be quite far off the truth. 
. , . But let’s do our best.) 


[* Mid-1970s' population figures in this printing .] 


The Isles of Earth 


267 


Table 17 The World Island * 

AREA 

(SQUARE MILES) POPULATION 

Asia 16,500,000 1,950,000,000 

Africa 1 1 ,500,000 336,000,000 

Europe 3,800,000 653,000,000 

The World Island 31,800000 2,939,000,000 

The World Island contains a little more than half the 
total land area of the globe. Even more significantly, it 
contains three-quarters of the Earth’s population, It has 
a fair claim to the name. 

The only tract of land that even faintly compares to 
the World Island in area and population is the American 
mainland, first discovered by primitive Asians many thou- 
sands of years ago, again by the Icelandic navigator Leif 
Ericsson in 1000 a.d., and finally by the Italian navigator 
Giovanni Caboto (John Cabot to the English nation, 
which employed his services) in 1497. ... I don’t men- 
tion Columbus because he discovered only islands prior 
to 1497. He did not touch the American mainland until 
1498. 

Columbus thought that the new mainland was part of 
Asia, and so indeed it might have been. Its complete 
physical independence of Asia was not demonstrated till 
1728, when the Danish navigator Vitus Bering (employed 
by the Russians) explored what is now known as the 
Bering Sea and sailed through what is now called the 
Bering Strait, to show that Siberia and Alaska were not 
connected. 

There is, therefore, a second immense island on Earth 
and this one is, traditionally, divided into two continents: 
North America and South America, These, however, if 
one ignores the man-made Panama Canal, are connected, 
and a man can travel from Alaska to Patagonia without 
crossing salt water. 

There is no convenient name for the combined con- 
tinents. It can be called “the Americas” but that makes 

[* This article first appeared in fune 1966. In the years since , popula- 
tions have, of course f increased. I have changed all the tables in this 
article, therefore, to reflect that increase, 1 



268 


NUMBERS AND THE EARTH 


use of a plural term for what is a single tract of land and 
I reject it for that reason. 

I would like to suggest a name of my own— ‘the New 
World Island.” This capitalizes on the common (if old- 
fashioned) phrase “the* New World" for the Americas. 
It also indicates the same sort of relationship between the 
World Island and the New World Island that there is 
between England and New England, or between York 
and New York. 

The vital statistics on the New World Island are pre- 
sented in Table 18. As you see, the New World Island 
has about half the area of the World Island, but only a 
little over one-sixth the population. 

Table 18 The New World Island 
area 

(square miles) population 

North America 9,385,000 321,000,000 

South America 7,035,000 190,000,000 

The New World Island 1 6,420,000 511 ,000,000 

There are two other tracts of land large enough to be 
considered continents, and one tract which is borderline 
and is usually considered too small to be a continent. These 
are, in order of decreasing area: Antarctica (counting its 
ice cap), Australia, and Greenland. 

Since Greenland is almost uninhabited, I would like 
(as a pure formality) to lump it in with the group of 
what we may call “continental islands just to get it om 
of the way. We can then turn to the bodies of land smaller 
than Greenland and concentrate on those as a group. 
Table 19 lists the data on the continental-islands. 

Table 19 The Continental-Islands 
area 

(square miles) population 

The World Island 31,800,000 2,939,000,000 

The New World Island 16,420,000 511,000,000 

Antarctica 5,100,000 

Australia 2,970,000 

Greenland 840,000 


12,550,000 

47,000 


The hies of Earth 


269 


The bodies of land that remain — all smaller than Green- 
land — are what we usually refer to when we speak of 
“islands.” From here on in, then, when I speak of “islands” 
in this essay, I mear bodies of land smaller than Green- 
land and entirely surrounded by the sea. 

There are many thousands of such islands and they 
represent a portion of the land surface of the globe that 
is by no means negligible. Altogether (as nearly as I can 
estimate it) the islands have a total area of about 2,500,- 
000 square miles — so that in combination they are con- 
tinental in size, having almost the area of Australia. The 
total population is about 400,000,000, which is even more 
clearly continental in size, being well above the total 
population of North America. 

Let's put it this way, one human being out of every 
ten lives on an island smaller than Greenland. 

There are some useful statistics we can bring up in 
connection with the islands. First and most obvious is the 
matter of area. The five largest islands in terms of area 
are listed in Table 20. 

Table 20 The Largest Islands * 

AREA 

(square miles) 

New Guinea 312,329 

Borneo 290,285 

Madagascar 230,035 

Baffin 201,600 

Sumatra 163,145 

The largest island. New Guinea, spreads out over an 
extreme length of 1,600 miles. If superimposed on the 
United States, it would stretch from New York to Denver. 
In area, it is fifteen per cent larger than Texas. It has 
the largest and tallest mountain range outside those on the 
World Island and the New World Island, and some of 
the most primitive people in the world. 

Two other islands of the first five are members of the 
same group as New Guinea, It, Borneo, and Sumatra are 

t* These days it is becoming more customary to use the geographic names 
used by the inhabitants . Borneo is really Kalimantan, for instance , 1 wilt , 
however, use the more familiar name in every case,} 



270 


NUMBERS AND THE EARTH 


The Isles of Earth 


271 


all part of what used to be called the “East Indies/’ an 
archipelago stretching across four thousand miles of ocean 
between Asia and Australia, and making up by far the 
largest island grouping in the w T orld. The archipelago has 
an area of nearly one million square miles, and thus con- 
tains about forty per cent of all the island area in the 
world. The archipelago bears a population of perhaps 
121,000,000 or about thirty per cent that of all the island 
population in the world. 

In a way, Madagascar is like an East Indian island 
displaced four thousand miles westward to the other end 
of the Indian Ocean. It has roughly the shape of Sumatra 
and in size is midway between Sumatra and Borneo. Even 
its native population is more closely akin to those of 
Southeast Asia than to those of nearby Africa. 

Only Baffin Island of the five giants falls outside this 
pattern. It is a member of the archipelago lying in the 
north of Canada. It is located between the mouth of 
Hudson Bay and the coast of Greenland. 


populated as Europe. It is \% times as densely populated 
as the Netherlands, Europe’s most thickly peopled nation. 
This is all the more remarkable since the Netherlands is 
highly industrialized and Java is largely agricultural. After 
all, one usually expects an industrialized area to support 
a larger population than an agricultural one would. (And, 
to be sure, the Netherlands’ standard of living is much 
higher than Java’s.) 

Lagging far behind the big three are four other islands, 
each with more than ten million population. These are 
given in Table 22. (Kyushu, by the way, is another of 
the Japanese islands.) 

Notice that the seven most populous islands are all in 
the Eastern Hemisphere, all lying off the World Island 
or between the World Island and Australia. The most 
populous island of the Western Hemisphere is again one 
which most Americans probably can’t name. It is His* 
paniola, the island on which Haiti and the Dominican 
Republic are located. It’s population is 9,000,000. 


Oddly enough, not one of the five largest islands is a 
giant in respect to population. There are three islands, 
indeed (not one of which is among the five largest), 
which, among them, contain well over half of all the island 
people in the world. The most populous is probably not 
known by name to very many Americans. It is Honshu 
and, before you register a blank, let me explain that it is 
the largest of the Japanese islands, the one on which 
Tokyo is located. 

The three islands are given in Table 21. 

Table 21 The Most Populous Islands 


AREA 

SQUARE MILES RANK POPULATION 


Honshu 91,278 

Java 48,504 

Great Britain 88,133 


6 83,000,000 

12 78,000,000 

7 56,000,000 


Java is easily the most densely populated of the large 
islands. (I say “large islands” in order to exclude islands 
such as Manhattan.) It has a density of 1,600 people per 
square mile which makes it just nine times as densely 


Table 22 The Moderately Populous Islands * 



AREA 

(SQUARE MILES) 

RANK 

POPULATION 

Sumatra 

163,145 

5 

20,000,000 

Formosa 

13,855 

34 

15,000,000 

Ceylon 

25,332 

24 

15,000,000 

Kyushu 

14,791 

31 

12,000,000 


One generally thinks of great powers as located on the 
continents. All but one in history among the continental 
great powers were located on the World Island. (The one 
exception is the United States.) 

The great exception to the rule of continentalism among 
the great powers is, of course, Great Britain, ** In more 
recent times, Japan proved another. In fact, Great Britain 
and Japan are the only island nations that have been 
completely independent throughout medieval and modern 
history. 


[* Formosa is more properly known as Taiwan ; Ceylon as Srt Lanka.] 
** Tm not going to distinguish between England , Great Britain , the 
United Kingdom, and the British Istes . I can if I want to, though , 
never you fear! 



272 


273 


NUMBERS AND THE EARTH 

Nowadays, however (unless I have miscounted and I 
am sure that if I have I will be quickly enlightened by a 
number of Gentle Readers), there are no less than thirty- 
one island nations; thirty-one independent nations, that is, 
whose territory is to be found on an island or group of 
islands, and who lack any significant base on either the 

World Island or the New World Island. 

One of these nations, Australia, is actually a continent- 
nation by ordinary convention, but I’ll include it here 
to be complete. The thirty-one island nations (including 
Australia) are listed in Table 23, in order of population.’ 

Some little explanatory points should accompany this 
table. First the discrepancy between Great Britain’s area 
as an island and as a nation is caused by the fact that 
as a nation it includes certain regions outside its home 
island, notably Northern Ireland. Indonesia includes most 
but not all the archipelago I previously referred to as the 

East Indies. , . 

Virtually all the island people are now part of inde- 
pendent island nations. The largest island area that is still 
colonial is Papua New Guinea, the eastern half of the 
island of New Guinea. It has an area of 182,700 square 
miles, a population of 2,300,000, and is administered by 
Australia. I frankly don’t know how to classify Puerto 
Rico. It is self-governing to a considerable extent but if it 
is counted as an American colony, I think it may qualify 
as the most populous (pop. 2,700,000) nonindependent 

island remaining. . , , 

As you see from Table 23, the most populous island 
nation is neither Japan nor Great Britain, but Indonesia. 
It is, in fact, the fifth-most-populous nation in the world. 
Only China, India, the Soviet Union, and the United 
States (all giants in area) are more populous than Indo- 
nesia. . 

The only island nations that occupy less than a single 
island are Haiti and the Dominican Republic (which 
share Hispaniola) and Ireland, where the six northeastern 
counties are still part of Great Britain. The only islan 
nation which has part of its islands belonging to nations 


[* When this article first appeared nine years ago , there were only 
twenty-one island nations. Since then, ten more islands or island groups 
have become independent , and 1 have prepared a revised table , with 
updated population figures , to reflect that fact .] 


The Isles of Earth 
Table 23 The Island Nations 


AREA 

(square miles) population 


Indonesia 

735,268 

121,000,000 

Japan 

142,726 

105,000,000 

Great Britain 

94,220 

56,000,000 

Philippines 

115,707 

37,000,000 

Formosa 

13,885 

15,000,000 

Australia 

2,971.021 

12,500,000 

Ceylon 

25,332 

12,500,000 

Cuba 

44,218 

8,500,000 

Madagascar 

230,035 

7,500,000 

Haiti 

10,714 

5.000,000 

Dominican Republic 

IS, 816 

4,000,000 

Ireland 

27,135 

3,000,000 

New Zealand 

103,376 

2,900,000 

Singapore 

224 

2,200,000 

Jamaica 

4,232 

2,000,000 

Trinidad 

1,980 

950,000 

Mauritius 

720 

900,000 

Cyprus 

3,572 

630,000 

Fiji 

7,055 

575,000 

Malta 

122 

330,000 

Comoro 

864 

300,000 

Barbados 

166 

275,000 

Bahrain 

256 

230,000 

Bahamas 

5,382 

215,000 

Iceland 

39,768 

210,000 

Western Samoa 

1,097 

150,000 

Maidive Islands 

115 

115,000 

Grenada 

133 

110,000 

Tonga 

269 

100,000 

Sao Tome 

372 

69,000 

Nauru 

8 

8,000 


based on some continent is Indonesia. Part of the island 
of Borneo (most of which is Indonesian) makes up a 
portion of the new nation of Malaysia, based on nearby 
Asia. The eastern half of New Guinea (the western half 
of which is Indonesian) belongs to Australia. 

There are eighteen cities within these island nations 
which contain one million or more people. These are 



274 


NUMBERS AND THE EARTH 


The Isles of Earth 


275 


listed, in order of decreasing population, in Table 24 — 
and I warn you that some of the figures are not particu- 
larly trustworthy* 

Table 24 The Island Cities 


CITY 

NATION 

POPULATION 

Tokyo 

Japan 

11,400,000 

London 

Great Britain 

8,200,000 

Djakarta 

Indonesia 

4,500,000 

Osaka 

Japan 

3,000,000 

Sydney 

Australia 

2,800,000 

Melbourne 

Australia 

2,400,000 

Yokohama 

Japan 

2,300,000 

Nagoya 

Japan 

2,000,000 

Taipei 

Formosa 

1,750,000 

Havana 

Cuba 

1,700,000 

Kyoto 

Japan 

1,400,000 

Kobe 

Japan 

1,300,000 

Manila 

Philippines 

1,300,000 

Surabaja 

Indonesia 

1,300,000 

Bandung 

Indonesia 

1,100,000 

Kitakyushu 

Japan 

1,050,000 

Sapporo 

Japan 

1,000,000 

Birmingham 

Great Britain 

1,000,000 


Of these, Tokyo is certainly remarkable since it may 
be the largest city in the world. I say “may be” because 
there is a second candidate for the post — Shanghai. Popu- 
lation statistics for the Chinese People’s Republic (Com- 
munist China) are shaky indeed but there is a possibility 
that the population of Shanghai — a continental city — may 
be as high as 11,000,000, though figures as low as 7,000,- 
000 are also given. 

New York City, the largest city on the New World 
Island, is no better than a good fourth, behind Tokyo, 
Shanghai, and Greater London. New York is located 
mostly on islands, of course. Only one of its boroughs, 
the Bronx, is indisputably on the mainland. Still it is not 
on an island nation in the same sense that Tokyo or 
London is. 

If we exclude New York as a doubtful case, then the 
largest island city in the Western Hemisphere, and the 


only one in that half of the world to have a population 
of over a million, is Havana. 

That leaves only one item. In restricting the discussion 
of islands to those which are surrounded by salt water, 
have we been forced to neglect any important fresh 
water islands? 

In terms of size (rather than population) there is only 
one that is worth mentioning. It is a river island that very 
few in the world (outside Brazil) can be aware of. It is 
the island of Marajo, which nestles like a huge basketball 
in the recess formed by the mouth of the Amazon River. 

It is one hundred miles across and has an area of 
fifteen thousand square miles. It is larger than Formosa 
and if it were counted among the true islands of the sea 
it would be among the top thirty islands of the world, 
which is certainly not bad for a river island. However, it 
is a low-lying piece of land, swampy, often flooded, and 
right on the equator. Hardly anyone lives there. 

Its mere existence, though, shows what a monster of a 
river the Amazon is. 





INDEX 


A 

Abacus, 60 
Abraham, 70, 192 
Aconcagua, Mount, 250 
Adelard of Bath, 60 
Aepyornis, 208-09 
Africa, 265, 266 
African elephant, 200, 205 
Albatross, 209 
Alcuin, 194 
Aleph numbers, 79, 82 
Alexander the Great, 102, 
191, 200 
Alexandria, 102 
Algebra, fundamental theorem 
of, lb8 

Algebraic numbers, 112 
Alios aurs, 207 
Amazon River, 275 
Amoeba, 156, 158, 159 
Andromeda Galaxy, 152, 153 
Animals, size of, 199-200#, 


Antarctic Ocean, 233, 234 
Arabic numerals, 15-17, 56, 
132-33 

Aral Sea, 240, 241 
Archimedes, 14, 54n. } 70, 90, 
91, 217 
death of, 90 
pi and, 92 
Arctic Ocean, 233 
Arithmetic, fundamental 
theorem of, 108 
Asa, 70 
Asia, 264, 265 
Asimov Series, 37 
Atlantic Ocean, 233, 234 
Autumnal equinox, 186 
Azores, 254 


B 

Babylonian number symbols, 
5 



280 


Index 


Index 


281 


Bacon, Roger, 178 
Bacteria, 158 
Badwater, 251 
Baffin Island, 269-70 
Baikal, Lake, 242 
Baluchitherium, 205 
Beast, number of the, 8 
Bells, 40-43 
Bering, Vitus, 267 
Bering Strait, 235 
Bible, large numbers in, 70 
pi and, 90 
week and, 184 
year numbering and, 190-91 
Big bang, 228 
Binary system, 19-205. 
Biological Science, 131 
Blue whale, 202-03 
Borneo, 269, 270, 273 
Boston, 247 
Brachiosaurus, 207 
Buoyancy, principle of, 90 


C 

Cabot, John, 267 
Caesar, Julius, 174-77, 187, 
188 

death of, 176, 192 
Calendar month, 173 
Campbell, John, 68 
Cantor, Georg, 80, 81 
Cardano, Girolamo, 119 
Cards, playing, 48, 49 

permutations of, 45-46, 55- 
56 

Carolingian Renaissance, 194 
Carstenz, Mount, 252 
Caspian Sea, 240, 241 
Catherine I of Russia, 122 
Celestial Mechanics , 188 
Cells, 158 

Ceulen, Ludolf von, 97 


Challenger, 255 
Challenger Deep, 255 
Change, ringing, 40-41 
Charlemagne, 136, 193, 194, 
195 

Charles River, 262 
Chicago, 248*. 

Chimborazo, Mount, 260-61 
Chomolungma, 248 
Christian Era, 193 
Christmas, 180, 190 
Christmas Eve, 182 
Cicero, 176 
Cipher, 15 

Circle, squaring the, 107 
Cleopatra, 176 
Coefficients, 111 
Columbus, Christopher, 267 
Compass, 105 
Complex numbers, 126-27 
Compound addition, 135 
Computer, 25-26, 32, 33, 34 
Continent, 263 
Continuous creation, 228 
Continuum, 79 
Cook, Mount, 252 
Copernicus, Nicolas, 188 
Council of Nicaea, 175 
Crocodiles, 207-08 
Currency, 140, 144, 145 


D 

Dapsang, 249 
Dark Age, 194 
Dase, Zacharias, 99 
Davis, Philip J., 67 
Day, 166 

beginning of, 181-83 
Day, Donald, 68, 68*. 
Dead Sea, 240-43 
Death Valley, 251 
Descartes, Rene, 64 


Diaz, Bartholomew, 264 
Dinosaurs, 207 
Dionysius Exiguus, 193 
Djaja, Mount, 252 n. 
Dollar, 144, 145 
Dominican Republic, 272 
Drake Passage, 236 
Duodecimal system, 10 


E 

38-40, 113, 122 
Earth, 255-56 
Earthlight, 168 
Easter, 172-73 
East Indies, 269-70 
Eggs, bird, 210 
Egyptian calendar, 173-74 
Eight-based system, 34-35 
Elbrus, Mount, 250, 251 
Electrons, 223 
Elements, 102 
Elephants, 200, 201 
Eleven-based systems, 30-31 
English language, 142-43 
ENIAC, 100 
Equations, 111-13 
Equatorial bulge, 256 
Era of Abraham, 192 
Ericsson, Leif, 267 
Euclid, 59, 102, 103 
Eudoxus, 174 

Euler, Leonhard, 88, 120, 122, 
123 

Eurafrasia, 266 
Eurasia, 265 
Europe, 264, 265 
Eusebius, 192 
Everest, George, 248 
Everest, Mount, 248, 253, 255 
256 , 260 

Exhaustion, method of, 92 
^fconent, 53 


F 

f( ). 122 

F act and Fancy, 87 
Factorial, 42 
Fairy ffies, 212, 213 
Ferdinand of Brunswick, 108 
Fermat's Last Theorem, 36-37 
Fibonacci, Leonardo, 56, 60, 
61 

Fibonacci numbers, 56-60 
Fish, large, 208-09 
Fractions, 76-77, 118 
Frederick I of Prussia, 23 
Frederick II of Prussia, 122 
French Revolutionary calen- 
dar, 186 
Fresh water, 238 
Freund, John E. t S3 


G 

Galactic year, 158 
Galaxies, 220-21 
Galileo, 64, 80 
Gamow, George, 70 
Gasherbrum, Mount, 249, 
250*, 

Gauss, Karl Friedrich, 106, 
108, 109, 118 
Gelon, 217 
Gematria , 7 

Geometric progressions, 95-97 
George I of Great Britain, 23 
Giant clam, 212, 213 
Giant squid, 212, 213 
Gillies, Donald B., 62 
Goby fish, 209-12, 213 
God win- Austen, Henry 
Haversham, 249 
God win- Austen, Mount, 249 
Goliath beetle, 212, 213 
Googol, 52-53 



282 


Index 


Index 


283 


Googolplex, 66 
Gosainthan, Mount, 249, 250 
Great Britain, 270-72 
Great Lakes, 241, 244-45 
Great Rift Valley, 242 
Great Salt Lake, 240, 241 
Greeks, geometry and, 101 
number symbols of, 5-6 
Greenland, 268, 269 
Gregorian calendar, 142, 176 
Gregory XIII, Pope, 176 


H 

Hadrian, 8 
Haiti, 272 
Halloween, 182 
Hannibal, 192, 200 
Havana, 275 
Hawaii, 253 
Hegira, 171 
Heinlein, Robert, 42 
Henry I of England, 136 
Henry II of England, 60 
Her mite, Charles, 113 
Herod, 193 
Hieron II, 90 

Hillary, Edmund Perdval, 250 
Hilo, 256 

Himalaya Mountains, 248, 

249 

Hindu number symbols, 15 
Hispaniola, 271 
Honshu, 270 
Hours, 183-84 
Hubble, Edwin Powell, 219- 
20 , 220 «. 

Hubble Radius, 227-28 
Hubble's constant, 220-21 
Hubble’s Law, 220 
Hummingbirds, 209, 210, 211 
Huygens, Christian, 64 


I 

i, 120-21, 122 
Ice, 238 

Imaginary numbers, 1 19-20#. 
Index to the Science Fiction 
Magazines, 68 
Indian Ocean, 236 
Indonesia, 272, 273, 274 
Infinity, 68-69#. 

Inland Seas, salt in, 240, 241 
Intercalary day, 170 
Intercalary month, 172 
Ipsus, battle of, 200 
Irrational numbers, 78-79, 98- 
99, 118 
Island, 262-63 


J 

Japan, 271 
Jason, 265 
Java, 270-71 

Jefferson, Thomas, 139-40 
Tenner, Edward, 188 
Jewish calendar, 172 
Jewish Mundane Era, 192 
Jewish numbering system, 6-7 
Jordan River, 242 
Julian calendar, 142, 175, 176 


K 

Kalimantan, 269 /j. 
Karakorum Mountains, 249 
Kasner, Edward, 52 
Kepler, Johann, 10 
Kilauea, 253, 256, 257 
Kilimanjaro, Mount, 250 
Kiwi, 210 
Kodiak bear, 206 
Komodo monitor, 208 


Kosciusko, Mount, 251 
Kosciusko, Thaddeus, 251 
Kronecker, Leopold, 80 


L 

Lambert, Johann Heinrich, 98 
Laplace, Pierre Simon de, 3 52, 
188 

Leap day, 175-76 
Leap year, 175-76 
Leatherback turtle, 208, 213 
Leibniz, Gottfried Wilhelm, 
23, 24, 25, 94-95 
Leibniz series, 95 
Length, measure of, 148-49 
Liber Abaci , 56, 60 
Light, motion of, 160-61 
Lindemann, Ferdinand, 113 
Linear equations, 117 
liouville, Joseph, 112 
Lobster, 212, 213 
Logarithms, 203-04 
Lord's Day, 184 
Lore of Large Numbers, The, 
67 

Ludolfs number, 97 
Ludwig the Pious, 194 
Lunar calendar, 168 
Lunar month, 166 
Lunar-solar calendar, 172 
Lunar year, 167 
Luther, Martin, 8 


M 

McKinley, William, 251 
McKinley, Mount, 250, 251 
Madagascar, 269-70 
Magellan, Ferdinand, 233 
Manhattan, 247, 248 
Marajo, 275 


Mariana Trench, 255 
Marius, Simon, 152 
Mark Antony, 193 
Martinique, 256 
Mass, measure of, 155-57 
Mathematics and the Imagina* 
tion } 52 

Matter- waves, 223-24 
Mauna Kea, 253, 256 
Mauna Loa, 253, 256 
Mayans, 10 
Measures, 133-34#. 

Mersenne, Marin, 62, 64, 65 
Mersenne primes, 62-63 
Messier, Charles, 152 
Method of least squares, 108 
Methusaleh, 166 
Metric system, 139-41, 147- 
48#. 

prefixes of, 148 
Midsummer day, 187 
Midsummer Night 1 s Dream , 
A, 187 

Millay, Edna St. Vincent, 101 
Mindanao Trench, 255, 259 
Mithraism, 188 
Moa, 209 

Modern Introduction to 
Mathematics , A, 83 
Mohammed, 8, 171 
Mohammedan calendar, 168 
Month, 166 
beginning of, 185 
Moon, phases of, 166-70 
Morris, Gouverneur, 144 
Mountains, highest, 248-49#. 
Mundane Era, 192 


N 

Napoleon, 8, 186, 188, 189 
Nassau Mountains, 252 
Natural numbers, 118 



284 


Index 


Index 


285 


Nebular hypothesis, 152 
N egative numbers, 118-19 
Neponset River, 262 
Nero, 8 
Neutron, 223 
New Guinea, 252, 269 
Newman, James, 52 
Newton, Isaac, 23, 90 
New World Island, 268 
New Year's Eve, 182 
New York City. 274 
Nine-based systems, 30 
Nine Tailors , 40, 42 
Noon, 183 

Norgay, Tenzing, 250 
Nuclei, atomic, 224 
Numbers, 105-06 
large, 70 

symbols of, 3-4 $., 12-13 
Numerology, 7 

O 

Observable Universe, 22 1 , 
225-27 

Ocean, 234, 235 

depth of, 237, 254-55 
gases in, 234 
size of, 233-34, 237 
solids in, 234 
volume of, 238 
Of Matters Great and Small , 
67/1- 

Olympiads, 191-92 
One-to-one correspondence, 
74 

One , Two, Three , Infinity, 70 
Oughtred, William, 87-88 

P 

Pacific Ocean, 233, 234 
Papua, New Guinea, 272 


Passover, 186 
Pele, 256 
Pelee, Mount, 256 
Pendulum clock, 64 
Perfect numbers, 63, 66 
Peter the Great, 23, 122 
Pi, 87-88$. 

algebraic numbers and, 
112-13 

irrationality of, 98 
value of, 93-94$. 

Piccard, Jacques, 255 
Pico Alto, 254 
Pike, Nicolas, 132-33 
Pike's Arithmetic* 132 
Pisa, 56 
Plato, 105 
Points, 77-79 
Polo, Marco, 48, 264 
Polynomial equations, 111, 
117-18 
Porus, 200 

Positive numbers, 119 
Prime numbers, 59, 62-63 
Proton, 223 

radius of, 225-26 
Protozoa, 158 
Pseudopod, 158 
Pteranodon, 209 
Ptolemy, 102 
Ptolemy Euergetes, 174 
Puerto Rico, 272 
Pyrrhus, 200 
Pythagoras, 36 

Q 

Quadratic equation, 120 

R 

Rainfall, 239 
Ramadan, 171 


Rational numbers, 78 
Real numbers, 78-79, 124 
Recher, 99 

Recreational Mathematic* 
Magazine t 58, 59/l, 62 
Red Sea, 242 
Relativity, Theory of, 220 
Repeating decimals, 96-97 
Reticulated python, 208 
Revelation, Book of, 7-8 
Rio de Janeiro, 70 
Roman(s), 90 

number symbols of, 8-10$., 
132-33 

Roman calendar, 174-75 
Roman Era, 192 
Roman numerals, 10, 132-33 
Rosetta stone, 188 
Rosh Hashonah, 186 
Rotifers, 212, 213, 214 


S 

Sabbath, 184 
St Pierre, 256 
Salamanders, 209, 213 
Sand-Reckoner , The, 217 
Saturn, 187 
Saturnalia, 187 
Sayers, Dorothy, 40 
Scalar quantities, 127 
Schmidt, Maarten, 222 
Seasons, 167 
Seleucid Era, 191 
Seleucus, 191 
Seven Seas, 233 
Shakespeare, William, 187 
Shanghai, 274 
Shanks, William, 99, 100 
Sharp, Abraham, 97-98 
Shrews, 202, 206, 210, 213 
Sirius, 174 


Sitter, Willem de, 220 
Skewes, S., 67 
Skewes* number, 67 
Sodom and Gomorrah, 242 
Solar calendar, 173 
Solar year, 173-74 
Solomon, 70 
Sosigenes, 175 
Sothic cycle, 174 
Sperm whale, 206, 213 
Standard measures, 136 
Straightedge, 104 
Sudirman Range, 252n. 
Suez Isthmus, 265 
Sumerians, 10 
Summer solstice, 186-87 
Sunrise, 181-82 
Sunset, 181-82 
Superior, Lake, 241 
Synodic month, 167 


T 

Ten-based systems, 30 
Ternary system, 29 
Time, measure of, 157-58 
T-numbers, 54-55 
Tokyo, 274 

Transcendental numbers, 113 
14 

Transfinite numbers, 82 
Tropical year, 174 
Twelve-based systems, 31-32 


U 

Universe, 218-19$. 

particle numbers in, 225 
Ural Mountains, 265 
Urban IV, Pope, 176 



286 


'nde x 


V 

Vector quantities, 127 
Vega, George, 99 
Vernal equinox, 175-76, 186 
Vieta, Francis, 94, 97 
Vigesimal system, 10 
Viruses, 158 
Vityaz, 2 55 


W 

Wallenstein, Albrecht von, 10 
Walsh, Don, 255 
Washington, George, 179 
Week, 184-85 
Weekend, 184 
Whale shark, 208, 213 
White shark, 208, 213 
Whitney, Josiah Dwight, 25 1 


Whitney, Mount, 250, 251 
William II of Germany, 8 
Wimsey, Lord Peter, 42 
Winter solstice, 187 
World Islands, 266-67 
World Ocean, 233 


Y 

Yardstick, 136, 137 
Year, 167-68 

beginning of, 185-86#. 
numbering of, 190-91#. 


Z 

Zama, battle of, 200 
Zerah, 70 
Zero, 15 


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